Pathology of submeasures and $F_{\sigma }$ ideals

Abstract

We address some phenomena about the interaction between lower semicontinuous submeasures on ${\mathbb {N}}$ and $F_{\sigma }$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_{\sigma }$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_{\sigma }$ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $F_{\sigma }$ ideals using sequences in Banach spaces.

1 Introduction

Mazur [14] showed that every $F_{\sigma }$ ideal on ${{\,\mathrm{{\mathbb {N}}}\,}}$ is of the form ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )=\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:\varphi (A)<\infty \}$ for some lower semicontinuous submeasure (lscsm) $\varphi$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$. Mazur’s proof may be thought as a canonical way to get a submeasure for an $F_{\sigma }$ ideal, given a representation as a countable union of closed sets. Mazur’s construction provides integer-valued submeasures, but it hides interesting properties of both, the ideal and the submeasure. For example, it does not show whether the ideal is tall or how close is the submeasure to be a measure, even in the case of summable ideals, which are ideals induced by measures.

Farah [4] introduced the degree of pathology to quantify the closeness of a lscsm to be a measure. A lscsm has degree 1, and is called nonpathological, when is the supremum of a family of measures. We say that an $F_{\sigma }$ ideal ${\mathcal {I}}$ is nonpathological if ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for some nonpathological lscsm $\varphi$.

We study pathological and nonpathological $F_{\sigma }$ ideals from several points of view. Pathological $F_{\sigma }$ ideals are those that whenever ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for some lscsm $\varphi$, the degree of pathology of $\varphi$ is infinite. We show a family of examples of this kind of ideals, based on an ideal defined by K. Mazur in [14], and also present a way to identify some lscsm $\varphi$ whose degree of pathology is infinite. This last condition is necessary for ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ to be pathological but it is unknown whether is sufficient. We use this criterion to show that the degree of pathology of the usual lscsm $\chi$ inducing the Solecki’s ideal ${\mathcal {S}}$ is equal to infinite. That complements Figueroa and Hrušák’s result showing that ${\mathcal {S}}$ is pathological [5]. We also prove that nonpathology is preserved downward by the Rudin–Keisler pre-order. We show that our example of a pathological ideal has a restriction which is Rudin–Keisler above the Solecki’s ideal. Some questions about pathology of submeasures, pathology of ideals and Rudin–Keisler and Katětov orders are stated.

The second aspect of our study is concerned with the class of tall ideals (those satisfying that for every infinite set A there is an infinite set $B\subseteq A$ in the ideal). Tall ideals have been extensively investigated (see for instance [9, 11, 18]). A very useful tool for the study of tall ideals is the Katětov pre-order $\le _K$. Among tall Borel ideals one which has played a pivotal role is the random ideal ${\mathcal {R}}$ generated by the cliques and independent sets of the random graph [10]. It is known that if ${\mathcal {R}}\le _K {\mathcal {I}}$, then ${\mathcal {I}}$ is tall. For a while it was conjectured that ${\mathcal {R}}$ was a $\le _K$-minimum among Borel tall ideals [11], this turned out to be false [6], in fact, there are $F_\sigma$ tall ideals which are not $\le _K$ above ${\mathcal {R}}$. We show that if $\varphi$ is a nonpathological lscsm of type $c_0$ and ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, then ${\mathcal {R}}\le _K{{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. The notion of a lscsm of type $c_0$ was motivated by the results presented in the last section. It is known that ${\mathcal {R}}\le _K {\mathcal {I}}$ implies that ${\mathcal {I}}$ has a Borel selector (i.e. the set B in the definition of tallness can be found in a Borel way from A) [6,7,8]. We give a partial answer to the question of whether every nonpathological $F_\sigma$ tall ideal has a Borel selector.

Finally, in the last section, following the ideas introduced in [2, 3], we show how to represent $F_{\sigma }$ ideals using sequences in a Banach space. Let ${\textbf {x}}={(x_n)}_n$ be a sequence in a Banach space X. We say that $\sum x_n$ is perfectly bounded, if there is $k>0$ such that for all $F\subset {{\,\mathrm{{\mathbb {N}}}\,}}$ finite, $\left\| \sum _{n\in F}x_n \right\| \le k$. Let

$$\begin{aligned} {\mathcal {B}}({\textbf {x}})=\left\{ A\subseteq {\mathbb {N}}: \sum _{n\in A} x_n \text { is perfectly bounded }\right\} . \end{aligned}$$

We will show that an $F_{\sigma }$ ideal is nonpathological if and only if it is of the form ${\mathcal {B}}({\textbf {x}})$ for some ${\textbf {x}}$. In particular, when the space is $c_0$ we get the notion of a lscsm of type $c_0$ mentioned before and we show that ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ is tall iff $(x_n)_n$ is weakly null.

2 Preliminaries

For an arbitrary set X and a cardinal number $\kappa$, we denote by $[X]^{\kappa }$ (respectively $[X]^{<\kappa }$) the set of the subsets of X having cardinality $\kappa$ (resp, having cardinality smaller than $\kappa$). We say that a collection ${\mathcal {A}}$ of subsets of a countable set X is analytic (resp. Borel), if ${\mathcal {A}}$ is analytic (resp. Borel) as a subset of the Cantor cube $2^X$ (identifying subsets of X with characteristic functions). The collections $[X]^\omega$, $[X]^{<\omega }$ and $[X]^k$ ($k\in \omega$) are endowed with the subspace topology as subsets of $2^X$. We refer the reader to [12] for all non explained descriptive set theoretic notions and notations.

An ideal ${\mathcal {I}}$ on a set X is a collection of subsets of X such that (i) $\emptyset \in {\mathcal {I}}$ and , (ii) If $A, B\in {\mathcal {I}}$, then $A\cup B\in {\mathcal {I}}$ and (iii) If $A\subseteq B$ and $B\in {\mathcal {I}}$, then $A\in {\mathcal {I}}$. Given an ideal ${\mathcal {I}}$ on X, the dual filter of ${\mathcal {I}}$, denoted ${\mathcal {I}}^*$, is the collection of all sets $X\setminus A$ with $A\in {\mathcal {I}}$. We denote by ${\mathcal {I}}^+$ the collection of all subsets of X which do not belong to ${\mathcal {I}}$. The ideal of all finite subsets of ${{\,\mathrm{{\mathbb {N}}}\,}}$ is denoted by ${{\,\mathrm{\mathsf FIN}\,}}$. We write $A\subseteq ^*B$ if $A\setminus B$ is finite. There is a vast literature about ideals on countable sets (see for instance the surveys [9] and [18]). Since the collection of finite subsets of X is a dense set in $2^X$, there are no ideals containing $[X]^{<\omega }$ which are closed as subsets of $2^X$. An ideal ${\mathcal {I}}$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$ is $F_\sigma$ if there is a countable collection of closed subsets ${\mathcal {K}}_n\subseteq 2^X$ such that ${\mathcal {I}}=\bigcup _n {\mathcal {K}}_n$. On the other hand, there are no $G_\delta$ ideals containing all finite sets. Thus the simplest Borel ideals (containing all finite sets) have complexity $F_\sigma$.

A family ${\mathcal {A}}$ (not necessarily an ideal) of subsets of X is tall, if every infinite subset of X contains an infinite subset that belongs to ${\mathcal {A}}$. A tall family ${\mathcal {A}}$ admits a Borel selector, if there is a Borel function $S: [X]^\omega \rightarrow [X]^\omega$ such that $S(E)\subseteq E$ and $S(E)\in {\mathcal {A}}$ for all E.

A coloring is a function $c:[X]^2\rightarrow 2$, where $[X]^2$ is the collection of two elements subsets of X. A set $H\subseteq X$ is c-homogeneous, if c is constant in $[H]^2$. We denote by $\hom (c)$ the collection of homogeneous sets and by Hom(c) the ideal generated by the c-homogeneous sets, that is, $A\in Hom(c)$ iff there are c-homogeneous sets $H_1, \cdots , H_n$ such that $A\subseteq H_1\cup \cdots \cup H_n$. Since the singletons are trivially c-homogeneous sets, $[X]^{<\omega }\subseteq Hom(c)$, for every coloring c. It is easy to check that Hom(c) is $F_\sigma$ and, by Ramsey’s theorem, Hom(c) is tall. For some colorings c, Hom(c) is trivial. For example, if c satisfies that there are no infinitely many maximal 0-homogeneous sets, say $H_1, \cdots , H_n$ are the maximal 0-homogeneous sets, then Hom(c) is trivial. In fact, let . Then necessarily $c\{x,y\}=1$ (otherwise there is i such that $x,y\in H_i$). That is, $L={{\,\mathrm{{\mathbb {N}}}\,}}\setminus (H_1\cup \cdots \cup H_n)$ is 1-homogeneous. Hence ${{\,\mathrm{{\mathbb {N}}}\,}}$ is the union of finitely many homogeneous sets. The collection of homogeneous sets is a typical example of a tall family that has a Borel selector (see [7]).

A function $\varphi :{\mathcal {P}}({{\,\mathrm{{\mathbb {N}}}\,}})\rightarrow [0,\infty ]$ is a lower semicontinuous submeasure (lscsm) if $\varphi (\emptyset )=0$, $\varphi (A)\le \varphi (A\cup B)\le \varphi (A)+\varphi (B)$, $\varphi (\{n\})<\infty$ for all n and $\varphi (A)=\lim _{n\rightarrow \infty }\varphi (A\cap \{0,1,\cdots , n\})$.

Three ideals associated to a lscsm are the following:

$$\begin{aligned} \begin{array}{lcl} {{\,\mathrm{\mathsf FIN}\,}}(\varphi )&{} =&{} \{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:\varphi (A)<\infty \}. \\ \text{ Exh }(\varphi )&{} = &{}\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}: \lim _{n\rightarrow \infty }\varphi (A\setminus \{0,1,\dots , n\})=0\}.\\ \text{ Sum }(\varphi ) &{} = &{} \{A\subseteq {\mathbb {N}}:\sum _{n\in A}\varphi (\{n\})<\infty \}. \end{array} \end{aligned}$$

Notice that $\text{ Sum }(\varphi )\subseteq \text{ Exh }(\varphi )\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. These ideals have been extensively investigated. The work of Farah [4] and Solecki [16] are two of the most important early works for the study of the ideals associated to submeasures.

An ideal ${\mathcal {I}}$ is a P-ideal if for every sequence $(A_n)_n$ of sets in ${\mathcal {I}}$ there is $A\in {\mathcal {I}}$ such that $A_n\setminus A$ is finite for all n. The following representation of analytic P-ideals is the most fundamental result about them. It says that any P-ideal is in a sense similar to a density ideal.

Theorem 2.1

(S. Solecki [16]) Let ${\mathcal {I}}$ be an analytic ideal on ${{\,\mathrm{{\mathbb {N}}}\,}}$. The following are equivalent:

• (i) ${\mathcal {I}}$ is a P-ideal.

• (ii) There is a lscsm $\varphi$ such that ${\mathcal {I}}=\text{ Exh }(\varphi )$. Moreover, there is such $\varphi$ bounded.

In particular, every analytic P-ideal is $F_{\sigma \delta }$. Moreover, ${\mathcal {I}}$ is an $F_\sigma$ P-ideal, if, and only if, there is a lscsm $\varphi$ such that ${\mathcal {I}}=\text{ Exh }(\varphi )= {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

2.1 Integer valued submeasures

$F_{\sigma }$ ideals are precisely the ideals of the form ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for some lscsm $\varphi$. We recall this result to point out that such $\varphi$ can have an extra property, which we use later in our discussion of pathology of submeasures.

We say that a lscsm $\varphi$ is integer-valued if it takes values in ${{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}$ and $\varphi ({{\,\mathrm{{\mathbb {N}}}\,}})=\infty$.

Theorem 2.2

(Mazur [14]) For each $F_{\sigma }$ ideal ${\mathcal {I}}$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$, there is an integer-valued lscsm $\varphi$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. Moreover, there is such $\varphi$ satisfying that $\varphi (\{n\})=1$ for all $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$.

Proof

We include a sketch in order to verify the last claim. Let $({\mathcal {K}}_n)_n$ be a collection of closed hereditary subsets of $2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$ such that ${\mathcal {K}}_n\subseteq {\mathcal {K}}_{n+1}$, $A\cup B\in {\mathcal {K}}_{n+1}$ for all $A,B\in {\mathcal {K}}_n$ and ${\mathcal {I}}=\bigcup _n {\mathcal {K}}_n$. We can assume that ${\mathcal {K}}_0=\{\emptyset \}$, and since $\{\{n\}:\;n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}\cup \{\emptyset \}$ is a closed subset of $2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$, we can assume that $\{n\}\in {\mathcal {K}}_1$ for all $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Then the submeasure associated to $({\mathcal {K}}_n)_n$ is given by $\varphi (A)=\min \{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; A\in {\mathcal {K}}_n\}$, if $\emptyset \ne A\in {\mathcal {I}}$, and $\varphi (A)=\infty$, otherwise. $\square$

We now highlight two relevant properties of integer-valued submeasures.

Proposition 2.3

Let $\varphi$ be an unbounded integer-valued lscsm such that $\varphi (\{x\})=1$ for all $x\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Then

• (i) Every $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$ belongs to the range of $\varphi$.

• (ii) For every integer $k\ge 2$, there is a finite set B such that $\varphi (B)=k$ and $\varphi (C)<k$ for all $C\subset B$ with $C\ne B$.

Proof

• (i) Suppose not and let $n=\min ({{\,\mathrm{{\mathbb {N}}}\,}}\setminus range(\varphi ))$. Since $\varphi ({{\,\mathrm{{\mathbb {N}}}\,}})=\infty$, by the lower semicontinuity there is a finite set A such that $\varphi (A)=\min \{\varphi (B): \;n<\varphi (B)\}$. Let $m=\varphi (A)$. We can also assume that A is $\subseteq$-minimal with that property, that is, $\varphi (B)<n$ for all $B\subseteq A$ with $B\ne A$. Let $x\in A$, then

  $$\begin{aligned} \varphi (A)\le \varphi (A\setminus \{x\})+\varphi (\{x\})<n+1\le m, \end{aligned}$$

  a contradiction.

• (ii) By (i) there is a $\subseteq$-minimal finite set B such that $\varphi (B)=k$.

$\square$

3 Pathology of submeasures

The ideal $\text{ Sum }(\varphi )$ is also induced by a measure determined by its values on singletons. Namely, $\mu _{\varphi }(A)=\sum _{n\in A}\varphi (\{n\})$ is a measure and $\text{ Sum }(\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\mu _{\varphi })=\text{ Exh }(\mu _{\varphi })$. These ideals are the most well-behaved among $F_{\sigma }$ or P-ideals. Several good properties of summable ideals are shared by the larger class of nonpathological ideals. We say that $\mu$ is dominated by $\varphi$, if $\mu (A)\le \varphi (A)$ for all A, in this case, we write $\mu \le \varphi$. A lscsm $\varphi$ is nonpathological if it is the supremmum of all ($\sigma$-additive) measures dominated by $\varphi$. A quite relevant application of this kind of submeasures was Farah’s proof that $\text{ Exh }(\varphi )$ has the Radon-Nikodým property if $\varphi$ is nonpathological [4]. So far, nonpathology has been considered a property only about submeasures, we extend it to ideals as follows.

Definition 3.1

We say that an $F_{\sigma }$ ideal ${\mathcal {I}}$ is nonpathological if there is a nonpathological lscsm $\varphi$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

We show examples of ideals induced by both, a pathological and a nonpathological submeasure. The following proposition gives us a criterion for showing that an integer-valued submeasure is pathological.

Proposition 3.2

Let $\varphi$ be an integer valued lscsm on a set X. Suppose there is a finite set $A\subseteq X$ with $|A|\ge 2$ such that $\varphi (A\setminus \{x\})<\varphi (A)$ for all $x\in A$ and $\varphi (A)<|A|$. Then $\varphi$ is pathological.

Proof

Suppose $\varphi$ is nonpathological. Let $m=|A|$ and $x_0\in A$ be such that $\varphi (A\setminus \{x_0\})=\max \{\varphi (A\setminus \{x\}):\; x\in A\}$. Since $\varphi$ takes integer values and $\varphi (A)/m<1$, $\varphi (A\setminus \{x_0\})+\varphi (A)/m < \varphi (A)$. Pick a measure $\mu \le \varphi$ such that

$$\begin{aligned} \varphi (A\setminus \{x_0\})+\varphi (A)/m\;<\mu (A)\le \varphi (A). \end{aligned}$$

There is $y\in A$ such that $\mu (\{y\})\le \varphi (A)/m$. Then

$$\begin{aligned} \mu (A\setminus \{y\})=\mu (A)-\mu (\{y\})\ge \mu (A)-\varphi (A)/m>\varphi (A\setminus \{x_0\}) \ge \varphi (A\setminus \{y\}), \end{aligned}$$

which contradicts that $\mu \le \varphi$. $\square$

Now we present a very elementary example of a pathological lscsm.

Example 3.3

Let $\varphi$ be the lscsm defined on $\{0,1,2\}$ by $\varphi (\emptyset )=0$, $\varphi (a)=1$ if $0<\vert a \vert \le 2$ and $\varphi (\{0,1,2\})=2$. Then $\varphi$ is the minimal example of a pathological integer-valued submeasure on a finite set, where singletons have submeasure 1.

By elementary reasons, all $F_{\sigma }$ ideals and all analytic P-ideals can be induced by a pathological lscsm, as the following proposition shows.

Proposition 3.4

Let $\varphi$ be any lscsm on ${{\,\mathrm{{\mathbb {N}}}\,}}$. There is a pathological lscsm $\psi$ such that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\psi )$ and $\text{ Exh }(\varphi )=\text{ Exh }(\psi )$.

Proof

Let $\varphi _0$ be the submeasure on $\{0,1,2\}$ defined in Example 3.3. Let

$$\begin{aligned} \psi (A)=\varphi _0(A\cap \{0,1,2\})+\varphi (A\setminus \{0,1,2\}). \end{aligned}$$

$\psi$ is pathological, as $\psi \restriction \{0,1,2\}=\varphi _0$. Clearly $\psi$ works.

Now we present an example of a pathological lscsm $\varphi$ and a nonpathological lscsm $\psi$ such that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\psi )$ and ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall.

Example 3.5

Let $(B_n)_n$ be a partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ into infinite sets. The ideal $\mathcal{E}\mathcal{D}$ is defined as the ideal generated by pieces and selectors of the partition $(B_n)_n$. Let ${\mathcal {K}}_0$ be the set $\{\emptyset \}$ and

$$\begin{aligned} {\mathcal {K}}_1= \{H\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}: H\subseteq B_n\;\text {for some }n\}\cup \{H\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}: H\text { is a partial selector for }(B_n)_n \}. \end{aligned}$$

Then ${\mathcal {K}}_1$ is closed hereditary and $\mathcal{E}\mathcal{D}$ is the ideal generated by ${\mathcal {K}}_1$. Let

$$\begin{aligned} {\mathcal {K}}_{n+1}= \{H_1\cup \cdots \cup H_{n+1}: H_i\in {\mathcal {K}}_n\; \text {for }1\le i\le n+1\}. \end{aligned}$$

Let $\varphi$ be the Mazur’s submeasure defined in the proof of Theorem 2.2 for this family of closed hereditary sets. Clearly $\mathcal{E}\mathcal{D}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. We use Proposition 3.2 to show that $\varphi$ is pathological. Pick a set $A=\{x_1, x_2, x_3\}$ such that $x_1\in B_0$ and $x_2, x_3\in B_1$. Notice that $\varphi (A)=2$ and $\varphi (A\setminus \{y\})=1$ for all $y\in A$.

Consider the submeasure on ${{\,\mathrm{{\mathbb {N}}}\,}}$ given by

$$\begin{aligned} \psi (A)=\min \{m\in {{\,\mathrm{{\mathbb {N}}}\,}}:(\forall n> m)\vert A\cap B_n\vert \le m\}. \end{aligned}$$

It is easy to see that ${{\,\mathrm{\mathsf FIN}\,}}(\psi )=\mathcal{E}\mathcal{D}$. Now, let us consider the family

$$\begin{aligned} {\mathcal {S}}=\{\mu _n^F: n\in {{\,\mathrm{{\mathbb {N}}}\,}}, F\in [B_n]^{n+1}\} \end{aligned}$$

where, for each n and F, $\mu _n^F$ is the counting measure supported on F. Note that $\psi (A)=\sup \{ \mu (A):\mu \in {\mathcal {S}}\}$, since for every A and every m the following conditions are equivalent:

• For all $n\ge m$, $\vert A\cap B_n\vert \le m$, and

• For all n, and all $F\in [B_n]^{n+1}$, $\mu _n^F(A)\le m$.

Hence, $\psi$ is nonpathological.

For the sake of completeness, we mention that both submeasures $\varphi$ and $\psi$ from the previous example remain pathological and nonpathological respectively, when they are restricted to $\Delta =\bigcup _n C_n$, where each $C_n$ is a fixed subset of $B_n$ with cardinality $n+1$. $\mathcal{E}\mathcal{D}_{fin}$ denotes the restriction of $\mathcal{E}\mathcal{D}$ to $\Delta$. It is immediate to see that, in general, every restriction of a nonpathological submeasure is nonpathological, while some restrictions of pathological submeasures are nonpathological.

3.1 Degrees of pathology

Farah’s approach to pathology of submeasures on ${{\,\mathrm{{\mathbb {N}}}\,}}$ includes a concept of degree of pathology [4]. Associated to each lscsm $\varphi$, there is another lscsm ${\widehat{\varphi }}$, defined as follows.

$$\begin{aligned} {\widehat{\varphi }}(A)=\sup \{\mu (A):\mu \text { is a measure dominated by }\varphi \}, \end{aligned}$$

for all $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$. Clearly ${\widehat{\varphi }}$ is the maximal nonpathological submeasure dominated by $\varphi$.

The degree of pathology, which measures how far is a submeasure from being nonpathological, is defined by

$$ \begin{aligned} P(\varphi )=\sup \left\{ \frac{\varphi (A)}{{\widehat{\varphi }}(A)}: {\widehat{\varphi }}(A)\ne 0 \; \& \; A\in {{\,\mathrm{\mathsf FIN}\,}}\right\} . \end{aligned}$$

Note that $P(\varphi )=1$ if and only if $\varphi$ is nonpathological. Moreover, if $P(\varphi )=N<\infty$ then ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is equal to ${{\,\mathrm{\mathsf FIN}\,}}(\psi )$ for some nonpathological submeasure $\psi$: In fact, we have that ${\widehat{\varphi }}\le \varphi \le N{\widehat{\varphi }}$, thus ${{\,\mathrm{\mathsf FIN}\,}}({\widehat{\varphi }})\supseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )\supseteq {{\,\mathrm{\mathsf FIN}\,}}(N{\widehat{\varphi }})$, and clearly ${{\,\mathrm{\mathsf FIN}\,}}({\widehat{\varphi }})= {{\,\mathrm{\mathsf FIN}\,}}(N{\widehat{\varphi }})$. Let us note that for every $F_{\sigma }$ ideal ${\mathcal {I}}$ and every $n>1$, there is a lscsm $\varphi$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and $P(\varphi )\ge n$. In fact, if ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi ')$ is nonpathological, we can modify $\varphi '$, as we did in the proof of Proposition 3.4, by inserting a copy of the submeasure $\psi _{2n}$ defined in Sect. 3.2.1. This proves that some nonpathological ideals may be defined by submeasures having arbitrarily large finite degrees of pathology. Moreover, if $\varphi '$ is an arbitrary (in particular for nonpathological) submeasure and A is an infinite set such that $\varphi '(A)<\infty$, we can modify $\varphi '$ by taking a submeasure $\psi$ on A with $P(\psi )=\infty$ and $\psi (A)=\varphi '(A)$, and define $\varphi (B)=\max \{\psi (B\cap A),\varphi '(B\setminus A)\}$. Hence ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\varphi ')$ but $P(\varphi )=\infty$. This construction proves that every $F_{\sigma }$ ideal properly containing ${{\,\mathrm{\mathsf FIN}\,}}$ is induced by a submeasure with infinite degree of pathology. However, this construction encapsulates the pathological part of the submeasure in a small set, what lefts the following question open.

Question 3.6

Is there a nonpathological $F_{\sigma }$ ideal ${\mathcal {I}}$ for which there is a lscsm $\varphi$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$, $P(\varphi )=\infty$ and $P(\varphi \upharpoonright A)<\infty$, for all $A\in {\mathcal {I}}$?

In light of the notion of degree of pathology, we can see that an $F_{\sigma }$ ideal ${\mathcal {I}}$ is pathological if and only if $P(\varphi )=\infty$ whenever $\varphi$ is a lscsm such that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={\mathcal {I}}$. In the next section we present an example of a pathological $F_{\sigma }$ ideal, as a particular case of a general method of constructing pathological $F_{\sigma }$ ideals.

3.2 Examples of pathological ideals

K. Mazur constructed an $F_{\sigma }$ ideal which is not contained in any summable ideal [14]. We show in this section that a variation of Mazur’s construction produces $F_{\sigma }$ ideals which are not contained in any nonpathological ideal.

We need the concept of a covering number similar to the one defined by J. Kelley [13]. Given a finite set K, a covering ${\mathcal {S}}$ of K and an element $i\in K$, B(i) denotes the cardinality of $\{s\in {\mathcal {S}}:i\in s\}$ and $m(K,{\mathcal {S}})$ denotes the minimum of all B(i) for $i\in K$. The covering number of ${\mathcal {S}}$ in K is defined by

$$\begin{aligned} \delta (K,{\mathcal {S}})=\frac{m(K,{\mathcal {S}})}{\vert \mathcal {S \vert }}. \end{aligned}$$

Lemma 3.7

Let K be a finite set and ${\mathcal {S}}$ a covering of K. If $\pi$ is a probability measure on K, then there is $s\in {\mathcal {S}}$ such that $\pi (s)\ge \delta (K,{\mathcal {S}})$.

Proof

Let us note that

$$\begin{aligned} \sum _{s\in {\mathcal {S}}} \pi (s)=\sum _{i\in K} B(i)\pi (\{i\})\ge m(K,{\mathcal {S}})\sum _{i\in K}\pi (\{i\})=m(K,{\mathcal {S}}). \end{aligned}$$

Therefore, there is $s\in {\mathcal {S}}$ such that $\pi (s)\ge \frac{m(K,{\mathcal {S}})}{\vert {\mathcal {S}}\vert }=\delta (K,{\mathcal {S}})$. $\square$

An interval partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ is a family $\{I_n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ of intervals of ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\min I_0=0$ and $\min I_{n+1}=\max I_n +1$. The next theorem is about submeasures defined using an interval partition, but it can be stated in terms of a family of pairwise disjoint finite sets which covers a given countable set.

Theorem 3.8

Let $\varphi$ be a lscsm on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that there is $M>0$ and an interval partition $\{I_n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ satisfying

• the family ${\mathcal {B}}=\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:\varphi (A)\le M\}$ covers ${{\,\mathrm{{\mathbb {N}}}\,}}$,

• $\sup _n\varphi (I_n)=\infty$, and

• $\varphi (B)=\sup \{\varphi (B\cap I_n):n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$, for all $B\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$.

Let ${\mathcal {S}}_n$ be a subfamily of ${\mathcal {P}}(I_n)\cap {\mathcal {B}}$ such that ${\mathcal {S}}_n$ covers $I_n$, $\epsilon _n=\delta (I_n,{\mathcal {S}}_n)$ and $\delta =\inf \{\epsilon _n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. If $\delta >0$, then

• (i) ${\widehat{\varphi }}$ is bounded,

• (ii) $P(\varphi )=\infty$,

• (iii) ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is not contained in any nontrivial nonpathological $F_{\sigma }$ ideal, and

• (iv) ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is a pathological $F_{\sigma }$ ideal.

Proof

Note that (ii) follows immediately from (i), and (iv) follows from (iii). Let us prove (i). Let $\mu$ be a finitely supported measure dominated by $\varphi$. Then there are:

• 1. a finite set $F\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$,

• 2. a probability measure $\pi _j$ on $I_j$, for each $j\in F$, and

• 3. $\lambda _j>0$ for each $j\in F$

such that the support of $\mu$ is contained in $\bigcup _{j\in F}I_j$ and $\mu \upharpoonright I_j=\lambda _j\pi _j$. By Lemma 3.7, there is $s_j$ in ${\mathcal {S}}_j$ such that $\pi _j(s_j)\ge \epsilon _j$. Then,

$$\begin{aligned} 0<\delta \sum _{j\in F} \lambda _j\le \sum _{j\in F}\lambda _j\epsilon _j\le \sum _{j\in F}\lambda _j\pi _j(s_j)=\mu \left( \bigcup _{j\in F}s_j\right) \le \varphi \left( \bigcup _{j\in F} s_j\right) \le M. \end{aligned}$$

Hence $\sum _{j\in F}\lambda _j\le \frac{M}{\delta }$. Since $\mu \left( \bigcup _{j\in F}I_j\right) =\sum _{j\in F}\lambda _j\pi _j(I_j)=\sum _{j\in F}\lambda _j$, it follows that $\mu$ is bounded by $\frac{M}{\delta }$. Since M and $\delta$ do not depend on $\mu$, we are done.

For (iii), we will prove that if $\psi =\sup {\mathcal {M}}$ for some family ${\mathcal {M}}$ of measures, and it is nontrivial in the sense that ${{\,\mathrm{\mathsf FIN}\,}}(\psi )\ne {\mathcal {P}}({{\,\mathrm{{\mathbb {N}}}\,}})$, then there is a set $B\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ which is not in ${{\,\mathrm{\mathsf FIN}\,}}(\psi )$. For each $n>0$, define $k_n\in {{\,\mathrm{{\mathbb {N}}}\,}}$ recursively: $k_0=0$ and

$$\begin{aligned} k_{n+1}=\min \left\{ j>k_n: \psi \left( \bigcup _{k_n\le i<j}I_j\right) >n\right\} . \end{aligned}$$

Notice that $k_{n+1}$ is well defined since $\bigcup _{j\ge m}I_j\not \in {{\,\mathrm{\mathsf FIN}\,}}(\psi )$ for all m. Let us pick a measure $\mu _n\in {\mathcal {R}}$ such that

$$\begin{aligned} \mu _n\left( \bigcup _{j=k_n}^{k_{n+1}-1}I_j\right) \ge n. \end{aligned}$$

By Lemma 3.7, for each $k_n\le j<k_{n+1}$, there is $s_j\in {\mathcal {S}}_j$ such that $(\mu _n\upharpoonright I_j)(s_j)\ge \delta \mu _n(I_j)$. Let us define $B_n=\bigcup _{j=k_n}^{k_{n+1}-1}s_j$. Hence,

$$\begin{aligned} \psi (B_n)\ge \mu _n\left( B_n\right) \ge \sum _{j=k_n}^{k_{n+1}-1}\delta \mu _n(I_j)=\delta \mu _n\left( \bigcup _{j=k_n}^{k_{n+1}-1}I_j\right) \ge \delta n. \end{aligned}$$

On the other hand, $\varphi (B_n)\le M$. Hence $B=\bigcup B_n$ is in ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and is not in ${{\,\mathrm{\mathsf FIN}\,}}(\psi )$. $\square$

3.2.1 Mazur’s example

We present an example of a pathological ideal using Theorem 3.8. Consider the following families of sets:

• Let $K_n$ be the set of all functions from n to $m=2n$, and

• ${\mathcal {S}}_n=\{{\hat{i}}:i=0,\dots , m-1\}$ where ${\hat{i}}=\{f\in K_n: i\notin range(f)\}$.

Note that for each $n>1$, ${\mathcal {S}}_n$ is a covering of $K_n$, and no subset of ${\mathcal {S}}_n$ with at most n sets is a covering of $K_n$. Moreover, every $f:n\rightarrow m$ avoids at least n values in m. That proves that $m(K_n,{\mathcal {S}}_n)=n$, and since $\vert {\mathcal {S}}_n \vert =m$, we have that $\epsilon _n=\delta (K_n,{\mathcal {S}}_n)=\frac{1}{2}$. Let $\psi _n$ be the subsmeasure on $K_n$ defined by

$$\begin{aligned} \psi _n(A)=\min \left\{ r:\exists b\in [m]^r A\subseteq \bigcup _{i\in b}{\hat{i}}\right\} , \end{aligned}$$

for all $A\subseteq K_n$. By identifying the sets $K_n$ with members of the corresponding interval partition, we can define the lscsm $\psi$ by

$$\begin{aligned} \psi (A)=\sup \{\psi _n(A\cap K_n):n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}. \end{aligned}$$

Hence, we have that $\psi (K_n)=n+1$ and the family ${\mathcal {B}}=\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:\psi (A)\le 1\}$ is a covering of ${{\,\mathrm{{\mathbb {N}}}\,}}$. By Theorem 3.8, ${\mathcal {M}}={{\,\mathrm{\mathsf FIN}\,}}(\psi )$ is a pathological ideal. For the sake of completeness, let us note that $P(\psi _n)=\frac{n+1}{2}$.

3.3 Submeasures with infinite pathological degree

In this section we present a sufficient condition for a submeasure to have infinite pathological degree.

Theorem 3.9

Let $\varphi$ be an unbounded lscsm on ${\mathbb {N}}$. Suppose there is $M>0$ such that the family

$$\begin{aligned} {\mathcal {B}}=\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:\varphi (A)\le M\} \end{aligned}$$

is a covering of ${{\,\mathrm{{\mathbb {N}}}\,}}$. Let $\{K_n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be a strictly increasing sequence of finite sets such that $\bigcup K_n={{\,\mathrm{{\mathbb {N}}}\,}}$ and ${\mathcal {S}}_n$ a subfamily of ${\mathcal {P}}(K_n)\cap {\mathcal {B}}$ such that ${\mathcal {S}}_n$ covers $K_n$. Let $\delta _n=\delta (K_n,{\mathcal {S}}_n)$ and $\delta =\inf \{\delta _n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. If $\delta >0$, then ${\widehat{\varphi }}$ is bounded and $P(\varphi )=\infty$.

Proof

Note that it is enough to prove that there is a uniform bound for all measures dominated by $\varphi$. Let $\mu$ be a finitely supported measure dominated by $\varphi$. Then, there are:

• 1. $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$,

• 2. a probability measure $\pi$ on $K_n$, and

• 3. $\lambda >0$

such that the support of $\mu$ is contained in $K_n$ and $\mu =\lambda \pi$. By Lemma 3.7, there is s in ${\mathcal {S}}_n$ such that $\pi (s)\ge \delta _n$. Thus,

$$\begin{aligned} 0<\lambda \delta \le \lambda \delta _n\le \lambda \pi (s)=\mu (s)\le \varphi (s)\le M. \end{aligned}$$

Hence $\lambda \le \frac{M}{\delta }$. Since $\mu (K_n)=\lambda \pi (K_n)=\lambda$, it follows that $\mu$ is bounded by $\frac{M}{\delta }$. Since M and $\delta$ do not depend from n, we are done. $\square$

We remark that the collection ${\mathcal {B}}$ mentioned above is a covering of ${{\,\mathrm{{\mathbb {N}}}\,}}$ iff $\{n\}\in {\mathcal {B}}$ for all $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$. This requirement is easy to satisfy, for instance it holds if ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall.

It remains open the following question:

Question 3.10

Let $\varphi$ be a lscsm on ${{\,\mathrm{{\mathbb {N}}}\,}}$ and $(K_n, {\mathcal {S}}_n)_n$ and ${\mathcal {B}}$ as in the hypothesis of Theorem 3.9. Suppose ${\widehat{\varphi }}$ is bounded, is $\delta >0$? If $\delta =0$, is ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ nonpathological?

The calculation of $\delta$ for a given submeasure may not be easy to do. In the next section we present two examples illustrating this computation. However, we can naturally associate sequences $(K_n)_n$ and $({\mathcal {S}}_n)_n$ to every lscsm given by Theorem 2.2 and thus, in principle, we can calculate the corresponding $\delta$.

3.3.1 Solecki’s ideal

The ideal ${\mathcal {S}}$ is defined [17] on the countable set $\Omega$ of all clopen subsets of the Cantor set $2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$ whose measure¹ is equal to $\frac{1}{2}$ and it is generated by the sets of the form $E_x=\{a\in \Omega : x\in a\}$ for $x\in 2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$. Hrušák’s Measure Dichotomy [10] establishes that any pathological analytic P-ideal has a restrictions to a positive set which is Katětov above ${\mathcal {S}}$. Solecki’s ideal is critical for the class of analytic P-ideals but is not a P-ideal. It is a tall $F_{\sigma }$ ideal but we do not know if it is pathological, although, we show below that there is a lscsm $\chi$ such that ${\mathcal {S}}={{\,\mathrm{\mathsf FIN}\,}}(\chi )$ with $P(\chi )=\infty$.

Consider a lscsm $\chi$ on $\Omega$ given by

$$\begin{aligned} \chi (A)=\min \{n: \;\exists {x_1}, \cdots , x_n\in 2^{{{\,\mathrm{{\mathbb {N}}}\,}}},\; \; A\subseteq E_{x_1}\cup \cdots \cup E_{x_n}\}. \end{aligned}$$

It is clear that ${\mathcal {S}}={{\,\mathrm{\mathsf FIN}\,}}(\chi )$. We will show that it satisfies the hypothesis of Theorem 3.9 and therefore $P(\chi )=\infty$.

Let us denote by $\langle s\rangle$ the clopen set $\{x\in 2^{{\mathbb {N}}}:s\subseteq x\}$, for $s\in 2^{<\omega }$. For every $n>1$, we define

$$\begin{aligned} \Omega _n=\{b\in \Omega : (\forall s\in 2^n)(\langle s\rangle \subseteq b \text { or } \langle s \rangle \cap b=\emptyset )\}. \end{aligned}$$

Notice that $\Omega _n$ is an increasing sequence of sets whose union is equal to $\Omega$: Let $a\in \Omega$, since a is a compact set, select a finite covering $\{\langle c_1\rangle ,\dots \langle c_r\rangle \}$ of a with $c_1,\dots c_r\in 2^{<\omega }$. Let n be the length of the longest $c_j$. Then $a\in \Omega _n$.

For a given $s\in 2^n$, let ${\tilde{s}}=\{b\in \Omega _n:\langle s\rangle \subseteq b\}$ and ${\mathcal {S}}_n$ be the family $\{{\tilde{s}}:s\in 2^n\}$. Note that $\chi ({\tilde{s}})=1$ for all s. Moreover, $\chi (A)=\min \{k:\exists s_1,\dots ,s_k\in 2^n: A\subseteq \bigcup _{i=1}^k\tilde{s_i}\}$, for all $A\subseteq \Omega _n$.

Note that for every $B\in [2^n]^{2^{n-1}}$, $\bigcup _{s\in B}{\tilde{s}}\ne \Omega _n$, while for every $C\in [2^n]^{2^{n-1}+1}$, $\bigcup _{s\in C}{\tilde{s}} = \Omega _n$. Thus $\chi (\Omega _n)=2^{n-1}+1$. Also note that each $a\in \Omega _n$ belongs exactly to $2^{n-1}$ many sets in ${\mathcal {S}}_n$. Thus, $\delta (\Omega _n,{\mathcal {S}}_n)=\frac{1}{2}$ for all n.

3.3.2 The ideal $\mathcal{E}\mathcal{D}_{fin}$

Recall that in Example 3.5 we show that $\mathcal{E}\mathcal{D}_{fin}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ where $\varphi$ is defined on $\Delta =\bigcup _n C_n$, where $C_n$ is a subset of $B_n$ of size $n+1$, and is given by $\varphi (A)=\sup \{\mu _n(A\cap C_n):n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ for $\mu _n$ the counting measure on $C_n$. Consider

• $K_n=\bigcup _{j\le n}C_n$,

• ${\mathcal {S}}_0=\{C_0\}$ and ${\mathcal {S}}_{n+1}=\{s\cup \{j\}:s\in {\mathcal {S}}_n \; \& \; j\in C_{n+1}\}$

for all n. Notice that $|{\mathcal {S}}_{n}|=(n+1)!$

In Example 3.5 we have shown that $\varphi$ is non pathological, i.e. $P(\varphi )=1$. On the other hand, we show next that $\delta _n=\delta (K_n,{\mathcal {S}}_n)=\frac{1}{n+1}$, for all $n\ge 1$. Let $k\le n$ and $i,j\in C_k$. It is easy to verify that $B(i)=B(j)=(n+1)!/k$, i.e. i belongs to as many elements of ${\mathcal {S}}_n$ as j does. Thus $\delta (K_n,{\mathcal {S}}_n)=\frac{1}{n+1}$.

3.4 Degrees of pathology and Rudin–Keisler order

For a given ideal ${\mathcal {J}}$ and a function $f:{{\,\mathrm{{\mathbb {N}}}\,}}\rightarrow {{\,\mathrm{{\mathbb {N}}}\,}}$, an ideal $f({\mathcal {J}})$ is defined as follows:

$$\begin{aligned} f({\mathcal {J}})=\{A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}:f^{-1}(A)\in {\mathcal {J}}\}. \end{aligned}$$

An ideal ${\mathcal {I}}$ is said to be Rudin–Keisler below ${\mathcal {J}}$ if ${\mathcal {I}}=f({\mathcal {J}})$ for some f, denoted ${\mathcal {I}}\le _{RK}{\mathcal {J}}$ and f is called a Rudin–Keisler reduction of ${\mathcal {J}}$ in ${\mathcal {I}}$. The Katětov order is defined by ${\mathcal {I}}\le _{K}{\mathcal {J}}$ if ${\mathcal {I}}\subseteq {\mathcal {I}}'$ for some ${\mathcal {I}}'\le _{RK}{\mathcal {J}}$. The analytic P-ideals have an elegant classification in the Katětov order, given by the Hrusak’s Measure Dichotomy (Theorem 4.1 in [10]). Such classification uses the degrees of pathology.

We now show how Rudin–Keisler order impacts on degrees of pathology.

Lemma 3.11

Let ${\mathcal {I}}$ and ${\mathcal {J}}$ be $F_{\sigma }$ ideals, $\varphi$ a lscsm such that ${\mathcal {J}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and f a Rudin–Keisler reduction of ${\mathcal {J}}$ in ${\mathcal {I}}$. Let $\varphi _f$ be defined by

$$\begin{aligned} \varphi _f(A)=\varphi (f^{-1}(A)) \end{aligned}$$

for all $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$. Then, the following hold.

• 1. $\varphi _f$ is a lscsm and ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi _f)$,

• 2. if $\varphi$ is a measure then $\varphi _f$ is also a measure,

• 3. if $\nu$ is a measure dominated by $\varphi$ then $\nu _f$ is a measure dominated by $\varphi _f$,

• 4. for all $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$, $\widehat{\varphi _f}(A)\ge {\widehat{\varphi }}(f^{-1}(A))$, and

• 5. $P(\varphi _f)\le P(\varphi )$.

Proof

(1)–(3) are routine. For (4), $\widehat{\varphi _f}(A)=\sup \{\nu (A):\nu \text { is a measure dominated by }\varphi _f\}$ $\ge \sup \{\mu _f(A):\mu \text { is a measure dominated by }\varphi \}=\sup \{\mu (f^{-1}(A)):\mu \text { is a measure}\text { dominated by }\varphi \}={\widehat{\varphi }}(f^{-1}(A))$, for all $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$. For (5) we use (4) to argue that for all $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$,

$$\begin{aligned} \frac{\varphi _f(A)}{\widehat{\varphi _f}(A)} \le \frac{\varphi (f^{-1}(A))}{{\widehat{\varphi }}(f^{-1}(A))}\le P(\varphi ). \end{aligned}$$

Hence, $P(\varphi _f)\le P(\varphi )$. $\square$

Theorem 3.12

Let ${\mathcal {I}}$ and ${\mathcal {J}}$ be $F_{\sigma }$ ideals such that ${\mathcal {I}}\le _{RK}{\mathcal {J}}$. If ${\mathcal {J}}$ is nonpathological then ${\mathcal {I}}$ is nonpathological.

Proof

It follows immediately from Lemma 3.11 and the fact that an ideal is non pathological if it has a submeasure with finite pathological degree. $\square$

Recently, Figueroa and Hrusak [5] proved that nonpathological $F_{\sigma }$ ideals (and every restriction of them) are Katetov-below ${\mathcal {Z}}$, the ideal of asymptotic density zero sets. This implies that ${\mathcal {S}}$ is pathological. However we do not know if ${\mathcal {S}}$ plays a critical role among pathological $F_{\sigma }$ ideals in Katetov or Rudin–Keisler orders.

Question 3.13

How is ${\mathcal {S}}$ related (Katětov, Rudin–Keisler) with pathological $F_{\sigma }$ ideals?

As we show next, Mazur’s ideal ${\mathcal {M}}$ defined in Sect. 3.2.1 confirms the critical role played by Solecki’s ideal ${\mathcal {S}}$ in the Katětov order for the collection of pathological ideals.

Theorem 3.14

There is an ${\mathcal {M}}$-positive set X such that ${\mathcal {S}}\le _{RK} {\mathcal {M}}\upharpoonright X$.

Proof

Recall that $K_n$ is the collection of all functions from n to 2n and ${\mathcal {S}}_n=\{{\hat{i}}:\;i=0,\dots , m-1\}$. Let $X_n$ ($n\ge 0$) be the set of all one-to-one functions in $K_n$, and $X=\bigcup _{n}X_{2^n}$. Note that X is ${\mathcal {M}}$-positive since $X\cap K_{2^n}$ cannot be covered by less or equal than $2^n$ members of ${\mathcal {S}}_{2^n}$, for all n. For each n, fix an enumeration $\{s^n_0,s^n_1, \dots ,s^n_{2^n-1}\}$ of $2^n$. The Rudin–Keisler reduction f from X to $\Omega$ works as follows. If $r\in X_{2^n}$ then

$$\begin{aligned} f(r)=2^{{{\,\mathrm{{\mathbb {N}}}\,}}}\setminus \bigcup _{j=0}^{2^n-1}\langle s^{n+1}_{r(j)}\rangle . \end{aligned}$$

Note that $f(r)\in \Omega _{n}$.

We show first, that if $A\in {\mathcal {S}}$ then $f^{-1}(A)\in {\mathcal {M}}\upharpoonright X$. Suppose $A=\{a\in \Omega :x\in a\}$ for some $x\in 2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$. Then for all n, $A\cap \Omega _{n}=\{a\in \Omega _n:\; \langle x\upharpoonright n\rangle \subseteq a\}$, but for some $j_n\in \{0,\dots ,2^{n}-1\}$ it happens that $x\upharpoonright n=s^{n}_{j_n}$. Hence $f^{-1}(A)\cap X_n$ is the set $\widehat{j_n}=\{r\in X_n: j_n\notin range(r)\}$, what proves that $\psi (f^{-1}(A))=1$.

On the other hand, suppose that $f^{-1}(A)\in {\mathcal {M}}$. Then there exists N such that for all n, there is $J_n\in [2^{n}]^{N}$ such that $r\cap J_n=\emptyset$ for all $r\in f^{-1}(A)\cap K_{2^n}$. Thus for all n there are $c^{n}_{j_1},c^{n}_{j_2},\dots ,c^{n}_{j_N}\in 2^{n}$ such that $\langle c^{n}_{j_i}\rangle \subseteq A\cap \Omega _n$, for some $i=1,\dots , N$. Hence, $\chi (f^{-1}(A))\le N$, what proves that $A\in {\mathcal {S}}$. $\square$

4 Tallness and Borel selectors

In this section we address the question of the tallness of ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. As a first remark, we notice that there is a simple characterization of when $\text{ Exh }(\varphi )$ is tall. In fact, $\text{ Exh }(\varphi )$ is tall iff $\text{ Sum }(\varphi )$ is tall iff $\varphi (\{n\})\rightarrow 0$. Indeed, if we let $C_{n}=\{x\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; 2^{-n}<\varphi (\{x\})\}$, then $\text{ Exh }(\varphi )$ is tall iff each $C_n$ is finite. Notice also that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, whenever $\text{ Exh }(\varphi )$ is tall, as $\text{ Exh }(\varphi )\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. But the converse is not true, that is, it is possible that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall while $\text{ Sum }(\varphi )$ and $\text{ Exh }(\varphi )$ are not (see Example 4.3).

On the other hand, Grebík and Hrušák [6] showed that there are no simple characterizations of the class of tall $F_{\sigma }$ ideals, in fact, they showed that the collection of closed subsets of $2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$ which generates an $F_\sigma$ tall ideal is not Borel as a subset of the hyperspace $K(2^{{{\,\mathrm{{\mathbb {N}}}\,}}})$.

4.1 Property A and ideals generated by homogeneous sets of a coloring

In this section we are going to examine two very different conditions implying tallness of an $F_{\sigma }$ ideal. We introduce a property weaker than requiring that $\lim _n \varphi (\{n\})=0$ but which suffices to get that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall.

Definition 4.1

A lscsm $\varphi$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$ has property A, if $\varphi ({{\,\mathrm{{\mathbb {N}}}\,}})=\infty$ and $\varphi (\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; \varphi (\{n\})>\varepsilon \})<\infty$ for all $\varepsilon >0$.

Property A can be seen as a condition about the convergence of $(\varphi (\{n\}))_n$ to 0, but in a weak sense. In fact, let us recall that for a given filter ${\mathcal {F}}$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$, a sequence $(r_n)_n$ of real numbers ${\mathcal {F}}$-converges to 0, if $\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}: |r_n|<\varepsilon \}\in {\mathcal {F}}$ for all $\varepsilon >0$. Let ${\mathcal {F}}$ be the dual filter of ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. Then $\varphi$ has property A iff $(\varphi (\{n\}))_n$ ${\mathcal {F}}$-converges to 0.

On the other hand, property A has also a different interpretation. We recall that an ideal ${\mathcal {I}}$ over ${{\,\mathrm{{\mathbb {N}}}\,}}$ is weakly selective [11], if given a positive set $A\in {\mathcal {I}}^+$ and a partition $(A_n)_n$ of A into sets in ${\mathcal {I}}$, there is $S\in {\mathcal {I}}^+$ such that $S\cap A_n$ has at most one point for each n. A submeasure $\varphi$ has property A, if ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ fails to be weakly selective in the following partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$: $A_{n+1}=\{x\in {{\,\mathrm{{\mathbb {N}}}\,}}: 1/2^{n+1} \le \varphi (\{x\}) < 1/2^n\}$ and $A_0= \{x\in {{\,\mathrm{{\mathbb {N}}}\,}}: 1\le \varphi (\{x\})\}$. In fact, any selector for $\{A_n:\; n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ belongs to $\text{ Exh }(\varphi )$ and thus to ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

Proposition 4.2

${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall for all lscsm $\varphi$ with property A.

Proof

Let $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ be an infinite set. If there is $\varepsilon >0$ such that $A\subseteq \{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; \varphi (\{n\})>\varepsilon \}$, then $A\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ as $\varphi$ has property A. Otherwise, pick $n_k\in A$ such that $\varphi (\{n_k\})\le 2^{-k}$ for all $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Let $B=\{n_k:\; k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. Then $\varphi (B)\le \sum _k \varphi (\{n_k\})<\infty$. $\square$

Note that any integer valued lscsm fails to have the property A. Thus, every $F_\sigma$ tall ideal ${\mathcal {I}}$ is induced by a lscsm $\psi$ without the property A (for example, the one given by the proof of Mazur’s theorem applied to ${\mathcal {I}}$).

Now we present a natural construction of submeasures with property A. In particular, it provides a lscsm $\varphi$ such that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall but $\varphi (\{n\})\not \rightarrow 0$.

Example 4.3

Let $\{B_n:n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be a partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ into infinite sets and $\{B_n^k:k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be a partition of $B_n$ satisfying:

• $B_n^0$ consists of the first $2^n(n+1)$ elements of $B_n$.

• $\min B_n^{k+1}=\min \{x\in B_n:x>\max B_n^k\}$.

• $\vert B_n^{k+1}\vert \ge \vert B_n^k\vert$.

Let $\nu _n^k$ be the measure on $B_n^k$ given by $\nu _n^k (\{x\})=\frac{n+1}{\vert B_n^k\vert }$ for all $x\in B_n^k$. Let

$$\begin{aligned} \varphi _n=\sup _k \nu _n^k \end{aligned}$$

and

$$\begin{aligned} \varphi =\sum _n \varphi _n. \end{aligned}$$

Then $\varphi$ is a nonpathological lscsm. We list some useful facts about this construction.

• 1. $\varphi (B_n)=\varphi (B_n^k)=n+1$ for all n and k.

• 2. Let $(n_i)_i$ and $(k_i)_i$ two sequences in ${{\,\mathrm{{\mathbb {N}}}\,}}$. Suppose $(n_i)_i$ is increasing. Then $\varphi (\bigcup _i B_{n_i}^{k_i})=\infty$.

• 3. $\varphi$ has property A. Let $\varepsilon >0$ and $M_{\varepsilon }=\{x\in {{\,\mathrm{{\mathbb {N}}}\,}}: \varphi (\{x\})\ge \varepsilon \}$. Notice that $\nu _n^k (\{x\})\le \frac{1}{2^n}$ for all $x\in B_n^k$. Let N be such that $2^{-N}<\varepsilon$, then $M_{\varepsilon }$ is disjoint from $B_{m}$ for all $m>N$ and thus $M_{\varepsilon }\subseteq B_0\cup \cdots \cup B_N$ belongs to ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

• 4. ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is not a P-ideal. In fact, the P-property fails at $(B_n)_n$. Indeed, let us suppose that $B_n\subseteq ^*X$ for all n. Then for each n there is $k_n$ such that $B_n^{k_n}\subseteq X$, and thus $\bigcup _nB_n^{k_n}\subseteq X$. By (2), $X\notin {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

• 5. Every selector of the $B_n$’s belongs to $\text{ Sum }(\varphi )$.

• 6. $B_n\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )\setminus \text{ Exh }(\varphi )$ for all n.

Now we present two particular examples of the previous general construction.

• (a) Suppose $\vert B_n^{k+1}\vert =\vert B_n^k\vert$ for all n and k. Notice that $\nu _n^k (\{x\})=\frac{n+1}{\vert B_n^k\vert }=1/2^n$ for all $x\in B_n$. Thus $\varphi (\{x\})=1/2^n$ for all $x\in B_n$ and $\lim _n \varphi (\{n\})$ does not exist. Then $\text{ Sum }(\varphi )$ and $\text{ Exh }(\varphi )$ are not tall, but ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall since it has the property A.

• (b) Suppose $\vert B_n^{k+1}\vert =\vert B_n^k\vert +n+1=(n+1)(2^n+k)$. Then $\nu _n^k (\{x\})=\frac{n+1}{\vert B_n^k\vert }=\frac{1}{2^n+k}$ for all $x\in B_n^k$. We show that $\varphi (\{m\})\rightarrow 0$, when $m\rightarrow \infty$. Given $\varepsilon >0$, we have seen that $M_{\varepsilon }=\{x\in {{\,\mathrm{{\mathbb {N}}}\,}}: \varphi (\{x\})\ge \varepsilon \}$ is disjoint from $B_{m}$ for all $m>N$ when $2^{-N}<\varepsilon$, and it is also disjoint from $B_n^k$ when $k^{-1}<\varepsilon$. Hence $M_{\varepsilon }$ is finite. We claim that $\text{ Sum }(\varphi )\ne \text{ Exh }(\varphi )\ne {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. By (4) and (6), it is sufficient to prove that there is $X\in \text{ Exh }(\varphi )\setminus \text{ Sum }(\varphi )$. For a fixed n, let $X=\{x_k:k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be such that $x_k\in B_n^k$. Since $\varphi (\{x_k\})=\frac{1}{2^n+k}$ for all k, $X\not \in \text{ Sum }(\varphi )$. On the other hand, $\varphi (\{x_k:\, k\ge m\})=\frac{1}{2^n+m}\rightarrow 0$ when $m\rightarrow \infty$.

In both examples (a) and (b), $\varphi$ is a nonpathological submeasure since it can be expressed as $\sup \{ \mu _s:s\in {\mathbb {N}}^{<\omega }\}$, where $\mu _s$ is defined by

$$\begin{aligned} \mu _s(D)= \sum _{j<\vert s \vert } \nu _{j}^{s(j)}(D\cap B_{j}^{s(j)}) \end{aligned}$$

for $s\in {\mathbb {N}}^{<\omega }$ and $D\subseteq {\mathbb {N}}$.

Now we present some examples of tall ideals which do not contain ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for any $\varphi$ with property A. Our examples are motivated by Ramsey’s theorem. We refer the reader to Sect. 2 where the notation is explained and to [7] for more information about this type of ideals

We say that a coloring $c:[X]^2\rightarrow 2$ favors color i, if there are no infinite $(1-i)$-homogeneous sets and in every set belonging to $Hom(c)^*$ there are $(1-i)$-homogeneous sets of any finite cardinality.

Proposition 4.4

Let $c:[X]^2\rightarrow 2$ be a coloring that favors a color. Then, Hom(c) does not contain ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for any lscsm $\varphi$ with property A.

Proof

Suppose c favors color 0. Let $\varphi$ be an arbitrary lscsm on X with property A and suppose that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )\subseteq Hom(c)$. We will construct a set A in $Hom(c)^+$ with $\varphi (A)<\infty$, which is a contradiction. Let $B_{n}=\{x\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; 2^{-n}<\varphi (\{x\})\}$ for each $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$. As $\varphi$ has property A, $B_n\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and ${{\,\mathrm{{\mathbb {N}}}\,}}\setminus B_n\in Hom(c)^*$. By hypothesis, for each $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$, there is a 1-homogeneous finite set $A_n$ with n elements and such that $A_n\cap B_n=\emptyset$. Since $\varphi (A_n)\le \frac{n}{2^ n}$, $A=\bigcup _n A_n\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. As ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )\subseteq Hom(c)$, there is a finite union of 0-homogeneous sets $C=C_1\cup \cdots \cup C_k$ such that $A\subseteq ^* C$. As A is infinite, there is $l>k$ such that $A_{l} \subseteq C$. Since $A_l$ has l elements, there are $i\le k$ and $x\ne y\in A_{l} \cap C_i$ which is imposible as $A_{l}$ is 1-homogeneous and $C_i$ is 0-homogeneous. A contradiction. $\square$

We present two examples of colorings satisfying the hypothesis of the previous proposition.

Example 4.5

Let $(P_n)_n$ be a partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $|P_n|=n$. Let c be the coloring given by $c\{x,y\}=0$ iff $x,y\in P_n$ for some n. This coloring favors color 1. Notice that $\mathcal{E}\mathcal{D}_{fin}$ is the ideal generated by the c-homogeneous sets.

Example 4.6

Let ${{\,\mathrm{{\mathbb {Q}}}\,}}$ be the rational numbers in [0, 1]. Let $\{r_n:\; n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be an enumeration of ${{\,\mathrm{{\mathbb {Q}}}\,}}$. The Sierpinski coloring, $c:[{{\,\mathrm{{\mathbb {Q}}}\,}}]^2\rightarrow 2$, is defined by $c\{r_n, r_m\}=0$ if $n<m$ iff $r_n<r_m$. Denote by $\mathcal{S}\mathcal{I}$ the ideal generated by the c-homogeneous sets. Observe that the homogeneous sets are exactly the monotone subsequences of $\{r_n:\; n\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. For each n, pick $X_n\subseteq (n,n+1)$ an infinite homogeneous set of color 0 and let $X=\bigcup _n X_n$. Then $X\in Hom(c)^+$. It is easy to check that $c\restriction X$ favors color 0.

To see that Proposition 4.4 is not an equivalence, we notice that $\mathcal{E}\mathcal{D}$ is the ideal generated by the homogeneous sets of a coloring not favoring any color, nevertheless, by an argument similar to that used in the proof of Proposition 4.4, it can be shown that $\mathcal{E}\mathcal{D}$ does not contain ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for any lscsm $\varphi$ with property A.

4.2 Katetov reduction to ${\mathcal {R}}$

Recall that ${\mathcal {R}}$ denotes the random ideal which is generated by the collection of cliques and independent sets of the random graph. It is easy to check that if ${\mathcal {R}}\le _K{\mathcal {I}}$, then ${\mathcal {I}}$ is tall. As we said before, for a while it was conjecture that any Borel tall ideal was Katetov above ${\mathcal {R}}$. This was shown to be false in [6]. In fact, they proved that there are $F_\sigma$ tall ideals which are not Katetov above the random ideal. In this section we present a partial answer to the question of whether ${\mathcal {R}}\le _K {\mathcal {I}}$ for a nonpathological tall ideal ${\mathcal {I}}$.

By the universal property of the random graph the following well known fact is straightforward.

Proposition 4.7

Let ${\mathcal {I}}$ be an ideal on ${{\,\mathrm{{\mathbb {N}}}\,}}$. Then, ${\mathcal {R}}\le _K {\mathcal {I}}$ if and only if there is $c:[{{\,\mathrm{{\mathbb {N}}}\,}}]^2\rightarrow 2$ such that $\hom (c)\subseteq {\mathcal {I}}$.

The next fact illustrates the previous observation.

Proposition 4.8

${\mathcal {R}}\le _K {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ for all lscsm $\varphi$ with property A.

Proof

Let

$$\begin{aligned} B_{k+1}=\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; 2^{-k-1}\;< \varphi (\{n\})\le 2^{-k}\} \end{aligned}$$

for $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$, $B_0=\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; \varphi (\{n\})>1\}$ and $B_{-1}=\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}: \varphi (\{n\})=0\}$. Let $c:[{{\,\mathrm{{\mathbb {N}}}\,}}]^{2}\rightarrow \{0,1\}$ be the coloring associated to the partition $\{B_k: \;k\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{-1\}\}$, that is to say, $c(\{n,m\})=1$ if, and only if, there is k such that $n,m\in B_k$. Then $A\in \hom (c)$ iff $A\subseteq B_{k}$ for some $k\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{-1\}$ or $|A\cap B_k|\le 1$ for all k. Notice that $B_{-1}\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and the argument used in the proof of Proposition 4.2 shows that $\hom (c)\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. $\square$

The next definition was motivated by the results presented in the last section about a representation of $F_{\sigma }$ ideals in Banach spaces.

Definition 4.9

A sequence of measures $(\mu _k)_k$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$ is of type $c_0$ if $\lim _k\mu _k(\{n\})=0$ for all n.

The main result of this section is the following:

Theorem 4.10

Let $\varphi =\sup _k\mu _k$ be a nonpathological lscsm where $(\mu _k)_k$ is a sequence of measures on ${{\,\mathrm{{\mathbb {N}}}\,}}$ of type $c_0$. The following are equivalent.

• (i) ${\mathcal {R}}\le _K {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

• (ii) ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall.

• (iii) $\lim _n\mu _k(\{n\})=0$ for all k and $\sup _n\varphi (\{n\})<\infty$.

For the proof we need some auxiliary results. Let $(\mu _k)_k$ be a sequence of measures on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\mu _k(\{n\})\le 1$ for all n and k. For $n,i\in {{\,\mathrm{{\mathbb {N}}}\,}}$, let

$$\begin{aligned} A^n_i=\left\{ k\in {{\,\mathrm{{\mathbb {N}}}\,}}: \; 2^{-i-1}< \mu _k(\{n\}) \le 2^{-i}\right\} \;\;\text{ and }\;\; A^n_\infty =\{k\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; \mu _k(\{n\})=0\}.\nonumber \\ \end{aligned}$$

(1)

Each ${\mathcal {P}}_n=\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$ is a partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ (allowing that some pieces of a partition are empty). Let $L_n=\{i\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; A^n_i\ne \emptyset \}$. We now show that, without loss of generality, we can assume that each $L_n$ is finite.

Lemma 4.11

Let $\varphi =\sup _k\mu _k$ be a nonpathological lscsm where $(\mu _k)_k$ is a sequence of measures on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\mu _k(\{n\})\le 1$ for all n and k. There is another sequence of measures $(\lambda _k)_k$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that

• (i) $\lambda _k\le \mu _k$ for all k.

• (ii) ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\psi )$ where $\psi =\sup _k \lambda _k$.

• (iii) If $({\mathcal {P}}_n)_n$ is the sequence of partitions associated to $(\lambda _k)_k$, then each $L_n$ is finite.

Proof

Let $\lambda _k(\{n\})=i/2^{n}$, if $i/2^{n} < \mu _k(\{n\})\le (i+1)/2^{n}$ for some $0< i< 2^n$, and $\lambda _k(\{n\})=0$ otherwise. Let $\psi =\sup _k\lambda _k$. Clearly $\lambda _k\le \mu _k$. Thus ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\psi )$. Notice that $|\mu _k(\{n\})-\lambda _k(\{n\})|\le 1/2^n$ for all n and k. Let $F\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ be a finite set. Then

$$\begin{aligned} \mu _k(F)= (\mu _k(F)-\lambda _k(F))+\lambda _k(F)\le \sum _{n\in F}1/2^n +\psi (F). \end{aligned}$$

Thus ${{\,\mathrm{\mathsf FIN}\,}}(\psi )\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. Let $\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$ be the sequence of partitions associate to $(\lambda _k)_k$. It is immediate that $A^n_i\ne \emptyset$, only if $i< n$. $\square$

Lemma 4.12

Let $\varphi =\sup _k\mu _k$ be a nonpathological lscsm where each $\mu _k$ is a measure on ${{\,\mathrm{{\mathbb {N}}}\,}}$. If ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, then $\lim _n\mu _k(\{n\})=0$ for all k and $\sup _n\varphi (\{n\})<\infty$.

Proof

For the first claim, fix $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$ and consider $B_i=\{n\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; \mu _k(\{n\})>1/i\}$ for $i\ge 1$. Suppose the conclusion is false. Then $B_i$ is infinite for some i. Since ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, there is $B\subseteq B_i$ infinite such that $\varphi (B)<\infty$. However, for every $F\subseteq B$ finite we have that $|F|/i< \mu _k(F)\le \varphi (F)\le \varphi (B)$, which contradicts that $\varphi (B)<\infty$.

For the second claim, suppose it is false and let $(n_k)_k$ be an increasing sequence in ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $k\le \varphi (\{n_k\})$. Clearly every infinite subset of $\{n_k:\; k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ does not belong to ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$, which contradicts the tallness of ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. $\square$

To each sequence of partitions ${\mathcal {P}}_n=\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$ of ${{\,\mathrm{{\mathbb {N}}}\,}}$ we associate a coloring $c:[{{\,\mathrm{{\mathbb {N}}}\,}}]^2\rightarrow 2$ as follows: for $n<m$

$$\begin{aligned} c\{n,m\} =1 \;\text {iff} \; ( \forall i\in {{\,\mathrm{{\mathbb {N}}}\,}}) \;\left( A^n_i\subseteq A^m_\infty \cup \bigcup _{j=i+1}^{\infty } A^m_j\right) . \end{aligned}$$

(2)

Lemma 4.13

Let $\varphi =\sup _k\mu _k$ be a nonpathological lscsm where $(\mu _k)_k$ is a sequence of measures on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\mu _k(\{n\})\le 1$ for all n and k. Let $\{A^n_i: \; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$, for $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$, be the sequence of partitions associated to $(\mu _k)_k$ and c be the associate coloring. Every c-homogeneous infinite set of color 1 belongs to ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

Proof

Let $H=\{n_i:\;i\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be the increasing enumeration of an infinite homogeneous set of color 1. We claim that for all k and m

$$\begin{aligned} \mu _k(\{n_0, \cdots , n_m\})=\sum _{i=0}^m \mu _k(\{n_i\})\le 2, \end{aligned}$$

which implies that $\varphi (H)\le 2$. In fact, fix k and m. We can assume without loss of generality that $\mu _k(\{n_i\})\ne 0$ for all $i\le m$. For each $i\le m$, let $j_i$ be such that $k\in A^{n_i}_{j_i}$. Since H is 1-homogeneous and $k\in A^{n_0}_{j_0}$ and $k\in A^{n_1}_{j_1}$, we have $j_0<j_1$. In general, we conclude that $j_0<j_1<\cdots <j_m$. Thus

$$\begin{aligned} \displaystyle \sum _{i=0}^m \mu _k(\{n_{i}\})\le \sum _{i=0}^\infty \frac{1}{2^{i}}. \end{aligned}$$

$\square$

Proof of Theorem 4.10

Clearly (i) implies (ii). Lemma 4.12 shows that (ii) implies (iii).

Suppose (iii) holds. Let $M=\sup _n\varphi (\{n\})$. Using $\mu _k/M$ instead of $\mu _k$ we can assume that $\mu _k(\{n\})\le 1$ for all n and k. By Lemma 4.11, we can also assume that the partition $\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$ associated to $(\mu _k)_k$ satisfies that $L_n=\{i\in {{\,\mathrm{{\mathbb {N}}}\,}}:\; A^n_i\ne \emptyset \}$ is finite for all n. Let c be the coloring given by (2). We will show that $\hom (c)\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$, then (i) follows by Proposition 4.7. By Lemma 4.13, it suffices to show that every infinite c-homogeneous set is of color 1. That is, we have to show that for all n, there is $m>n$ such that $c\{n,l\}=1$ for all $l\ge m$.

Fix $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Since $(\mu _k)_k$ is of type $c_0$, $A^n_i$ is finite for all $i\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Consider the following set:

$$\begin{aligned} S=\{k\in {{\,\mathrm{{\mathbb {N}}}\,}}: \; \mu _k(\{n\})\ne 0\}=\bigcup \{A^n_i: i\in L_n\}. \end{aligned}$$

As $L_n$ is finite, S is also finite. So, let $i_0$ be such that $2^{-i_0}<\mu _k(\{n\})$ for all $k\in S$. Since $\lim _m\mu _k(\{m\})=0$ for all k, there is $m>n$ such that

$$\begin{aligned} (\forall l\ge m)(\forall k\in S)(\; \mu _k(\{l\})<2^{-i_0}\;). \end{aligned}$$

That is to say, for all $k\in S$ and all $l\ge m$, there $j>i_0$ so that $k\in A^l_j$. Thus $c\{n,l\}=1$. $\square$

4.3 Borel selectors

Recall that a tall family ${\mathcal {I}}$ admits a Borel selector, if there is a Borel function $F: [{{\,\mathrm{{\mathbb {N}}}\,}}]^\omega \rightarrow [{{\,\mathrm{{\mathbb {N}}}\,}}]^\omega$ such that $F(A)\subseteq A$ and $F(A)\in {\mathcal {I}}$ for all A. This notion was studied in [6,7,8]. In this section we analyze Borel selectors for nonpathological tall $F_{\sigma }$ ideals.

The typical examples of families with Borel selectors are the collection of homogeneous sets. For instance, ${\mathcal {R}}$ admits a Borel selector and thus if ${\mathcal {R}}\le _K {\mathcal {I}}$, then ${\mathcal {I}}$ has a Borel selector. More generally, notice that if ${\mathcal {I}}\le _K {\mathcal {J}}$ and ${\mathcal {I}}$ has a Borel selector, then so does ${\mathcal {J}}$. The collection of all $K\in K(2^{{{\,\mathrm{{\mathbb {N}}}\,}}})$ such that the ideal generated by K has a Borel selector is a $\varvec{\Sigma }^1_2$ (see [7]), so it has the same complexity as the collection of codes for $F_{\sigma }$ ideals which are Katětov above ${\mathcal {R}}$ [6]. We do not know an example of an $F_{\sigma }$ ideal admitting a Borel selector which is not Katětov above ${\mathcal {R}}$. For concrete examples of $F_\sigma$ tall ideals without Borel selectors see [8].

Let ${\mathcal {Q}}_n$, for each $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$, be a collection of pairwise disjoint subsets ${{\,\mathrm{{\mathbb {N}}}\,}}$, say ${\mathcal {Q}}_n=\{B^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. The sequence $({\mathcal {Q}}_n)_n$ is eventually disjoint, if there is $p\in {{\,\mathrm{{\mathbb {N}}}\,}}$ such that for all $n\ne m$

$$\begin{aligned} \,B^n_i \cap B^m_i=\emptyset ,\;\;\text{ for } \text{ all } i>p. \end{aligned}$$

(3)

Theorem 4.14

Let $\varphi =\sup _k\mu _k$ where each $\mu _k$ is a measure on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\mu _k(\{n\})\le 1$ for all n and k. Suppose the sequence of partitions associate to $(\mu _k)_k$ is eventually disjoint. If ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, then it has a Borel selector.

Proof

We recall that the Schreier barrier is the following collection of finite subsets of ${{\,\mathrm{{\mathbb {N}}}\,}}$:

$$\begin{aligned} {\mathcal {S}}=\{s\in {{\,\mathrm{\mathsf FIN}\,}}: \; |s|=\min (s)+1\}. \end{aligned}$$

By the well known theorem of Nash-Williams [15], any coloring $c:{\mathcal {S}}\rightarrow 2$ has homogeneous sets. More precisely, for any $N\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ infinite, there is $M\subseteq N$ infinite such that for some $i\in \{0,1\}$, $c(s)=i$ for every $s\in [M]^{<\omega }\cap {\mathcal {S}}$. As usual, we denote by $\hom (c)$ the collection of all c-homogeneous sets. Moreover, $\hom (c)$ admits a Borel selector ( [7, Corollary 3.8]). We will find a coloring c of ${\mathcal {S}}$ such that $\hom (c)\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ and thus ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ also has a Borel selector.

Let $p\in {{\,\mathrm{{\mathbb {N}}}\,}}$ as in (3). Consider the following coloring ${\mathcal {S}}$:

$$\begin{aligned} c(\{q,n_1, \cdots , n_q\})=1\;\;\; \text{ iff } \text{ there } \text{ is } k\in {{\,\mathrm{{\mathbb {N}}}\,}} \text{ such } \text{ that } \mu _k(\{n_j\})\ge 2^{-p-1} \text{ for } \text{ all } 1\le j\le q\text{. } \end{aligned}$$

We will show that $hom(c)\subseteq {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

We first show that every infinite homogeneous set is of color 0. In fact, let A be an infinite set. It suffices to find $s\subseteq A$ with $s\in {\mathcal {S}}$ of color 0. Since ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, we also assume that $\varphi (A)<\infty$. Let $q\in A$ be such that $q2^{-p-1}>\varphi (A)$. Let $q<n_1<\cdots <n_q$ in A, we claim that $c(\{q,n_1, \cdots , n_q\})=0$. In fact, suppose not, and let k be such that $\mu _k(\{n_j\})\ge 2^{-p-1}$ for all $1\le j\le q$. Thus

$$\begin{aligned} \varphi (A)\ge \mu _k(\{n_1, \cdots , n_q\})=\sum _{j=1}^q\mu _k(\{n_j\})\ge q 2^{-p-1}>\varphi (A), \end{aligned}$$

a contradiction.

To finish the proof, it suffices to show that every infinite homogeneous set H of color 0 belongs to ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. Recall the definition of the partition ${\mathcal {P}}_n$ associated to $(\mu _k)_k$: ${\mathcal {P}}_n=\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}\}$ where each piece $A^n_i$ is defined by (1).

Let $q=\min (H)$ and $F\subseteq H\setminus \{q\}$ be a finite set. For each $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$, let

$$\begin{aligned} F_k=\{n\in F:\;\mu _k(\{n\})\ge 2^{-p-1}\}. \end{aligned}$$

Fix $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Since H is homogeneous of color 0, $|F_k|<q$. Let $n,m\in F\setminus F_k$ with $n\ne m$. Then $\mu _k(\{n\})<2^{-p-1}$ and $\mu _k(\{m\})<2^{-p-1}$. Thus $k\in A^n_i\cap A^m_j$ for some $i,j> p$ and by (3), we have that $i\ne j$. Thus

$$\begin{aligned} \sum _{n\in F\setminus F_k}\mu _k(\{n\})\le \sum _{i>p} 1/2^i. \end{aligned}$$

Then

$$\begin{aligned} \mu _k(F)=\sum _{n\in F}\mu _k(\{n\})= \sum _{n\in F_k}\mu _k(\{n\})+\sum _{n\in F\setminus F_k}\mu _k(\{n\}) \le q+2. \end{aligned}$$

As this holds for every k, we have that $\varphi (H\setminus \{q\})\le q+2$. Thus $H\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. $\square$

Proposition 4.15

Let ${\mathcal {Q}}_n$, for each $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$, be a collection of pairwise disjoint subsets of ${{\,\mathrm{{\mathbb {N}}}\,}}$, say ${\mathcal {Q}}_n=\{B^n_i:\;i\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$. Let $L_n=\{i\in {{\,\mathrm{{\mathbb {N}}}\,}}:\, B^n_i\ne \emptyset \}$. Suppose there is $l\in {{\,\mathrm{{\mathbb {N}}}\,}}$ such that $|L_n|\le l$ for every n. Then, there is an infinite set $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ such that $({\mathcal {Q}}_n)_{n\in A}$ is eventually disjoint.

Proof

By induction on l. Suppose $l=1$. For each n let $i_n$ be such that ${\mathcal {Q}}_n=\{B^n_{i_n}\}$. We consider two cases: (a) There is $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ infinite such that $(i_n)_{n\in A}$ is constant. Then $({\mathcal {Q}}_n)_{n\in A}$ is eventually disjoint. (b) There is $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ such that $i\in A\mapsto i_n$ is 1–1, then $({\mathcal {Q}}_n)_{n\in A}$ is eventually disjoint.

Suppose it holds for any sequence of pairwise disjoint sets with at most l non empty sets. Let ${\mathcal {Q}}_n=\{B^n_i:\;i\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ be such that $|L_n|\le l+1$. Let $i_n=\min (L_n)$. There are two cases to be considered. (a) Suppose $\sup _n i_n=\infty$. Pick $A=\{n_k:\; k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}$ such that $\max (L_{n_k})<\min (L_{n_{k+1}})$. Then $({\mathcal {Q}}_{n_k})_{k\in {{\,\mathrm{{\mathbb {N}}}\,}}}$ is eventually disjoint. (b) There is $B\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ infinite such that $(i_n)_{n\in B}$ is constant. Let ${\mathcal {Q}}_n'={\mathcal {Q}}_n\setminus \{B^n_{i_n}\}$. Then, we can apply the inductive hypothesis to $({\mathcal {Q}}_n')_{n\in B}$ and find $A\subseteq B$ infinite such that $({\mathcal {Q}}_n')_{n\in A}$ is eventually disjoint. Then $({\mathcal {Q}}_n)_{n\in A}$ is also eventually disjoint. $\square$

From the previous result and Theorem  4.14 we have

Corollary 4.16

Let $\varphi =\sup _k\mu _k$ where each $\mu _k$ is a measure on ${{\,\mathrm{{\mathbb {N}}}\,}}$ such that $\mu _k(\{n\})\le 1$ for all n and k. Let $\{A^n_i:\; i\in {{\,\mathrm{{\mathbb {N}}}\,}}\}\cup \{A^n_\infty \}$ be the associated partitions and $L_n=\{i\in {{\,\mathrm{{\mathbb {N}}}\,}}: A^n_i\ne \emptyset \}$. Suppose there is l such that $|L_n|\le l$ for all n. If ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )$ is tall, then it has a Borel selector.

The main open questions we have left are the following:

Question 4.17

Let ${\mathcal {I}}$ be nonpathological $F_\sigma$ tall ideal. Does ${\mathcal {R}}\le _K {\mathcal {I}}$? Does ${\mathcal {I}}$ have a Borel selector?

5 ${\mathcal {B}}$-representable ideals

In this section we show how to represent $F_{\sigma }$ ideals in a Banach space following the ideas introduced in [2, 3]. We first start with the representation in Polish abelian groups and later in Banach spaces.

Let $(G,+,d)$ be a Polish abelian group. We emphasis that the metric d is part of the definition. In fact the ideal associated to the group depends on the metric used. Let ${\textbf {x}}= (x_n)_n$ be a sequence in G. We say that $\sum _n x_n$ is unconditionally convergent, if there is $a\in G$ such that $x_{\pi (0)}+x_{\pi (1)}+\cdots + x_{\pi (n)} \rightarrow a$ for every permutation $\pi$ of ${{\,\mathrm{{\mathbb {N}}}\,}}$. We say that ${\textbf {x}}$ is perfectly bounded, if there exists $k>0$ such that for every $F\in {{\,\mathrm{\mathsf FIN}\,}}$

$$\begin{aligned} d\left( 0,\sum _{n\in F}x_n \right) \le k. \end{aligned}$$

We introduce two ideals. Given ${\textbf {x}}= (x_n)_n$ a sequence in G, let

$$\begin{aligned} {\mathcal {C}}({\textbf {x}})=\left\{ A\subseteq {\mathbb {N}}: \sum _{n\in A} x_n \text { is unconditionally convergent}\right\} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {B}}({\textbf {x}})=\left\{ A\subseteq {\mathbb {N}}: \sum _{n\in A} x_n \text { is perfectly bounded }\right\} . \end{aligned}$$

We observe that the ideal ${{\,\mathrm{{\mathcal {B}}}\,}}(x)$ depends on the metric of the group, not just on the topology, as is the case for ${{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$.

An ideal ${\mathcal {I}}$ is Polish ${\mathcal {B}}$-representable (resp. Polish ${{\,\mathrm{{\mathcal {C}}}\,}}$-representable) if there exists a Polish abelian group G and a sequence ${\textbf {x}}=(x_n)_n$ in G such that ${\mathcal {I}}={\mathcal {B}}({\textbf {x}})$ (resp. ${\mathcal {I}}={{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$). Polish ${{\,\mathrm{{\mathcal {C}}}\,}}$-representable ideals were studied in [2]. As a consequence of Solecki’s Theorem 2.1 they proved the following.

Theorem 5.1

(Borodulin et al [2]) An ideal ${\mathcal {I}}$ is Polish ${{\,\mathrm{{\mathcal {C}}}\,}}$-representable iff it is an analytic P-ideal.

5.1 ${{\,\mathrm{{\mathcal {B}}}\,}}$-representability in Polish groups

Let G be a Polish abelian group, d a complete, translation invariant metric on G and ${\textbf {x}}=(x_n)_n$ a sequence in G. We associate to ${\textbf {x}}$ a lscsm $\varphi _{\textbf {x}}$ as follows: $\varphi _{\textbf {x}}(\varnothing )=0$ and if $A\ne \varnothing$ we let

$$\begin{aligned} \varphi _{\textbf {x}}(A)=\sup \left\{ d\left( 0,\sum _{n\in F}x_n \right) : \varnothing \ne F \in [A]^{< \omega } \right\} . \end{aligned}$$

We show that $\varphi _{\textbf {x}}$ is indeed a lscsm. From its definition, it is clear that $\varphi _{\textbf {x}}(A)=\lim _{n\rightarrow \infty }\varphi _{\textbf {x}}(A\cap \{0,1,\cdots , n\})$ for every $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$. Let A and B be finite disjoint subsets of ${{\,\mathrm{{\mathbb {N}}}\,}}$. Then, by the translation invariance of d, we have

$$\begin{aligned} d\left( 0,\sum _{n\in A\cup B}x_n\right)= & {} d\left( 0, \sum _{n\in A}x_n+\sum _{n\in B}x_n\right) \le d\left( 0,\sum _{n\in A}x_n\right) +d\left( \sum _{n\in A}x_n,\sum _{n\in A}x_n+\sum _{n\in B}x_n\right) \\= & {} d\left( 0,\sum _{n\in A}x_n\right) +d\left( 0,\sum _{n\in B}x_n\right) . \end{aligned}$$

Let A, B be arbitrary subsets of ${{\,\mathrm{{\mathbb {N}}}\,}}$, and $\varepsilon >0$. Take a finite subset F of $A\cup B$ such that $d(0,\sum _{n\in F}x_n)\ge \varphi _{\textbf {x}}(A\cup B)-\varepsilon$. Since $d(0,\sum _{n\in F}x_n)\le d(0,\sum _{n\in F\cap A}x_n)+d(0,\sum _{n\in F\setminus A}x_n)\le \varphi _{\textbf {x}}(A)+\varphi _{\textbf {x}}(B)$, it follows that $\varphi _{\textbf {x}}$ is subadditive and thus is a lscsm.

Lemma 5.2

Let G be a Polish abelian group and ${\textbf {x}}=(x_n)_n$ a sequence in G. Then ${\mathcal {B}}({\textbf {x }})={{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$ and ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ is $F_{\sigma }$.

Proof

Let $A\in {\mathcal {B}}({\textbf {x}})$. Then there exists $k>0$ such that for every $\varnothing \ne F\in [A]^{<\omega }$,

$$\begin{aligned} d\left( 0, \sum _{n\in F}x_n \right) \le k. \end{aligned}$$

By the definition of $\varphi _{\textbf {x}}$, we have $\varphi _{\textbf {x}}(A)\le k$. Hence $A\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$. Conversely, assume that $A\in {{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$, then there exists $k>0$ such that $\varphi _{\textbf {x}}(A)\le k$. By the definition of $\varphi _{\textbf {x}}$, we clearly have that $A\in {\mathcal {B}}({\textbf {x }})$. Thus ${\mathcal {B}}({\textbf {x }})={{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$. Finally, since ${{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$ is $F_\sigma$ (see Theorem 2.2), so is ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$. $\square$

Theorem 5.3

The following statements are equivalent for any ideal ${\mathcal {I}}$ on ${{\,\mathrm{{\mathbb {N}}}\,}}$.

• (i) ${\mathcal {I}}$ is $F_\sigma$.

• (ii) ${\mathcal {I}}$ is ${{\,\mathrm{{\mathcal {B}}}\,}}$-representable in $({{\,\mathrm{\mathsf FIN}\,}},d)$ for some compatible metric d on ${{\,\mathrm{\mathsf FIN}\,}}$ (as discrete topological group).

• (iii) ${\mathcal {I}}$ is Polish ${{\,\mathrm{{\mathcal {B}}}\,}}$-representable.

Proof

By Lemma 5.2, (i) follows from (iii) and clearly (ii) implies (iii). To see that (i) implies (ii), let ${\mathcal {I}}$ be a $F_{\sigma }$ ideal on ${{\,\mathrm{{\mathbb {N}}}\,}}$. By Theorem 2.2, there is a lscsm $\varphi$ taking values on ${{\,\mathrm{{\mathbb {N}}}\,}}\cup \{\infty \}$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$. Then $\text{ Exh }{(\varphi )}={{\,\mathrm{\mathsf FIN}\,}}$. From the proof of Solecki’s theorem 2.1 we know that the complete metric on ${{\,\mathrm{\mathsf FIN}\,}}$ given by $d(A,B)=\varphi (A\triangle B)$ is compatible with the group structure of ${{\,\mathrm{\mathsf FIN}\,}}$. Let $x_n=\{n\}$ and ${\textbf {x}}=(x_n)_n$. We claim that ${\mathcal {I}}={{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ in the Polish group $({{\,\mathrm{\mathsf FIN}\,}}, d)$. First note that for every $\emptyset \ne F\in {{\,\mathrm{\mathsf FIN}\,}}$, $F=\sum _{n\in F}x_n$. Thus

$$\begin{aligned} d\left( \varnothing , \sum _{n\in F}x_n \right) =\varphi (F). \end{aligned}$$

By the lower semicontinuity of $\varphi$, we conclude that $\varphi =\varphi _{\textbf {x}}$. Therefore, by Lemma 5.2, ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$. $\square$

Notice that the proof of previous result shows that what matter for this type of representation is the translation invariant metric used on ${{\,\mathrm{\mathsf FIN}\,}}$, the topology is irrelevant as it can be assumed to be the discrete topology.

5.2 ${{\,\mathrm{{\mathcal {B}}}\,}}$-representability in Banach spaces

The motivating example of Polish representability is when the group is a Banach space. We rephrase the definitions of ${{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$ and ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ for the context of a Banach space. Let ${\textbf {x}}=(x_n)_n$ be a sequence in X.

• $\sum x_n$ converges unconditionally, if $\sum x_{\pi (n)}$ converges for all permutation $\pi :{{\,\mathrm{{\mathbb {N}}}\,}}\rightarrow {{\,\mathrm{{\mathbb {N}}}\,}}$.

• $\sum x_n$ is perfectly bounded, if there is $k>0$ such that for all $F\subset {{\,\mathrm{{\mathbb {N}}}\,}}$ finite, $\left\| \sum _{n\in F}x_n \right\| \le k$.

• The lscsm associated to ${\textbf {x}}$ is given by $\varphi _{\textbf {x}}(\emptyset )=0$ and for $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ non empty, we put

  $$\begin{aligned} \varphi _{\textbf {x}}(A)=\sup \left\{ \Vert \sum _{n\in F}x_n\Vert :\; F\subseteq A\; \text{ is } \text{ finite } \text{ non } \text{ empty }\right\} . \end{aligned}$$

  (4)

A motivation for studying ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ comes from the next result (part (iii) was not explicitly included but follows from the proof Theorem 1.3 of [3]).

Theorem 5.4

(Drewnowski-Labuda [3]) Let X be a Banach space. The following statements are equivalent:

• (i) X does not contain an isomorphic copy of $c_0$.

• (ii) ${\mathcal {C}}({{\textbf {x}}})$ is $F_\sigma$ in $2^{{{\,\mathrm{{\mathbb {N}}}\,}}}$ for each sequence ${{\textbf {x}}}$ in X.

• (iii) ${\mathcal {C}}({{\textbf {x}}})={\mathcal {B}}({{\textbf {x}}})$ for each sequence ${{\textbf {x}}}$ in X.

When working in Banach spaces, Theorem 5.1 is strengthened as follows.

Theorem 5.5

(Borodulin et al [2]) Let ${\mathcal {I}}$ be an ideal on ${{\,\mathrm{{\mathbb {N}}}\,}}$. The following are equivalent:

• (i) ${\mathcal {I}}=\text{ Exh }(\varphi )$ for a nonpathological lscsm $\varphi$.

• (ii) ${\mathcal {I}}={{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$ for some sequence ${\textbf {x}}=(x_n)_n$ in a Banach space.

The proof of the previous result also provides a characterization of ${{\,\mathrm{{\mathcal {B}}}\,}}$-representability on Banach spaces, as we show below. Since $l_\infty$ contains isometric copies of all separable Banach spaces, we have the following (already used in [2] for ${{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$).

Proposition 5.6

Let ${\textbf {x}}=(x_n)_n$ be a sequence in a Banach space X. There is ${\textbf {y}}=(y_n)_n$ in $l_\infty$ such that ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})={{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {y}})$.

From now on, we only work with $l_\infty$ (or $c_0$), this assumption implies that $\varphi _{\textbf {x}}$ has the important extra feature of being nonpathological.

It is convenient to have that the vectors $x_n\in l_\infty$ used in the representation of an ideal are of non negative terms. The following result was proved in [2] for ${{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$, a similar argument works also for ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$.

Lemma 5.7

Let ${\textbf {x}}=(x_n)_n$ be a sequence in $l_\infty$. Let ${\textbf {x'}}=(x'_n)$, where $x'_n(k)=|x_n(k)|$ for each $n,k\in {{\,\mathrm{{\mathbb {N}}}\,}}$, then ${\mathcal {B}}({\textbf {x}})={\mathcal {B}}({\textbf {x'}})$.

Now we recall why the lscsm $\varphi _{\textbf {x}}$ given by (4) is nonpathological when working on $l_\infty$. Let ${{\textbf {x}}}=(x_n)_n$ be a sequence in $l_\infty$ and assume that $x_n(k)\ge 0$ for all $n,k\in {{\,\mathrm{{\mathbb {N}}}\,}}$. Define a sequence of measures as follows. For $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ and $k\in {{\,\mathrm{{\mathbb {N}}}\,}}$, let

$$\begin{aligned} \mu _k(A)=\sum _{n\in A}x_n(k). \end{aligned}$$

Let $\psi =\sup _k \mu _k$, thus $\psi$ is a nonpathological lscsm. Notice that $\psi (\{n\})= \Vert x_n\Vert _\infty$ for all n and, more generally, for each $F\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ finite we have

$$\begin{aligned} \psi (F)=\left\| \sum _{n\in F}x_n \right\| _\infty . \end{aligned}$$

Since $\psi$ is monotone, $\psi (F)=\varphi _{{\textbf {x}}} (F)$ for every finite set F. Therefore $\psi =\varphi _{{\textbf {x}}}$.

Theorem 5.8

Let ${{\textbf {x}}}=(x_n)_n$ be a sequence in $l_\infty$ with $x_n\ge 0$ for all n. Then $\varphi _{{\textbf {x}}}$ is a nonpathological and

• (i) ${\mathcal {C}}({{\textbf {x}}})=\text{ Exh }(\varphi _{{\textbf {x}}})$.

• (ii) ${\mathcal {B}}({{\textbf {x}}})={{\,\mathrm{\mathsf FIN}\,}}(\varphi _{{\textbf {x}}})$.

Proof

(i) follows from the proof of Theorem 4.4 of [2]. (ii) Follows from Lemma 5.2. $\square$

Conversely, given a nonpathological lscsm $\varphi$, say $\varphi =\sup _k \mu _k$, where $\mu _k$ is a measure for each k, we associate to it a sequence ${{\textbf {x}}}_\varphi =(x_n)_n$ of elements of $l_\infty$ as follows: Given $n\in {{\,\mathrm{{\mathbb {N}}}\,}}$, let

$$\begin{aligned} x_n=(\mu _0(\{n\}),\ldots ,\mu _k(\{n\}),\ldots ). \end{aligned}$$

Notice that $\Vert x_n\Vert _\infty = \varphi (\{n\})$ for all n. For each $F\in {{\,\mathrm{\mathsf FIN}\,}}$, we have

$$\begin{aligned} \varphi (F)= & {} \sup \{\mu _k(F): k\in {{\,\mathrm{{\mathbb {N}}}\,}}\}\!=\!\sup \left\{ \sum _{n\in F}\mu _k(\{n\}): \;k\in {{\,\mathrm{{\mathbb {N}}}\,}}\right\} \!=\!\sup \left\{ \sum _{n\in F}x_n(k): \;k\in {{\,\mathrm{{\mathbb {N}}}\,}}\right\} \\= & {} \left\| \sum _{n\in F}x_n \right\| _\infty . \end{aligned}$$

In other words, $\varphi =\varphi _{{\textbf {x}}_\varphi }$. Part (i) of the following result is [2, Theorem 4.4] and (ii) follows from the above discussion.

Theorem 5.9

Let $\varphi$ be a nonpathological lscsm. Then

• (i) ${\mathcal {C}}({{\textbf {x}}}_\varphi )=\text{ Exh }(\varphi )$.

• (ii) ${\mathcal {B}}({{\textbf {x}}}_\varphi )={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

The following theorem is analogous to Theorem 5.5 but for ${{\,\mathrm{{\mathcal {B}}}\,}}$-representability.

Theorem 5.10

An ideal ${\mathcal {I}}$ is ${\mathcal {B}}$-representable in a Banach space if, and only if, there is a nonpathological lscsm $\varphi$ such that ${\mathcal {I}}={{\,\mathrm{\mathsf FIN}\,}}(\varphi )$.

Proof

Suppose ${\mathcal {I}}$ is a ${\mathcal {B}}$-representable ideal in a Banach space. By Lemma 5.6, we can assume that ${\mathcal {I}}$ is ${\mathcal {B}}$-representable in $l_\infty$. Let ${{\textbf {x}}}=(x_n)_n$ be a sequence in $l_\infty$ such that ${\mathcal {I}}={\mathcal {B}}({{\textbf {x}}})$. By Lemma 5.7 we assume that $x_n\ge 0$ for all n. Now the result follows from the previous considerations where we have shown that ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})={{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$. Conversely, if $\varphi$ is nonpathological, we have shown above that ${{\,\mathrm{\mathsf FIN}\,}}(\varphi )={{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}}_\varphi )$. $\square$

To end this section, we present an example of an ideal which is ${\mathcal {B}}$-representable in $c_0$ and is not a P-ideal, in particular, is not ${\mathcal {C}}$-representable in any Polish group.

Example 5.11

${{\,\mathrm{\mathsf FIN}\,}}\times \{\varnothing \}$ is ${\mathcal {B}}$-representable on $c_0$. Recall that ${{\,\mathrm{\mathsf FIN}\,}}\times \{\emptyset \}$ is defined by letting $A\in {{\,\mathrm{\mathsf FIN}\,}}\times \{\varnothing \}$ iff there is k such that $A\subseteq B_0\cup \cdots \cup B_k$, where $(B_n)_n$ is a partition of ${{\,\mathrm{{\mathbb {N}}}\,}}$ into infinitely many infinite sets. It is well known, and easy to verify, that ${{\,\mathrm{\mathsf FIN}\,}}\times \{\varnothing \}$ is not a P-ideal. Let ${{\textbf {x}}}=(x_n)_n$ be given by $x_n=me_n$, for $n\in B_m$, where $(e_n)_n$ is the usual base for $c_0$. It is easy to verify that ${{\,\mathrm{\mathsf FIN}\,}}\times \{\varnothing \}={\mathcal {B}}({{\textbf {x}}})$.

5.3 Tallness of ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$

It is easy to check that ${{\,\mathrm{{\mathcal {C}}}\,}}({\textbf {x}})$ is tall iff $\Vert x_n\Vert \rightarrow 0$. We show that the tallness of ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ is related to the weak topology. A classical characterization of perfect boundedness is as follows (see, for instance, [1, Lemma 2.4.6]).

Proposition 5.12

A series $\sum _nx_n$ in a Banach space is perfectly bounded iff $\sum _n |x^*(x_n)|<\infty$ for all $x^*\in X^*$.

From this we get the following.

Proposition 5.13

Let ${{\textbf {x}}}=(x_n)_n$ be a sequence in a Banach space X. If ${\mathcal {B}}({\textbf {x}})$ is tall, then ${\textbf {x}}=(x_n)_n$ is weakly null.

Proof

Suppose ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ is tall and $(x_n)_n$ is not weakly null. Then there is $A\subseteq {{\,\mathrm{{\mathbb {N}}}\,}}$ infinite and $x^*\in X^*$ such that $\inf _{n\in A}x^*(x_n)>0$. Let $B\subseteq A$ infinite such that $\sum _{n\in B}x_n$ is perfectly bounded. This contradicts proposition 5.12. $\square$

Thus, for a sequence ${\textbf {x}}=(x_n)_n$, we have the following implications:

$$\begin{aligned} (x_n)_n \; \text{ is } \Vert \cdot \Vert \text{-null } \;\Rightarrow {{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})\; \text{ is } \text{ tall }\;\Rightarrow (x_n)_n\; \text{ is } \text{ weakly } \text{ null }. \end{aligned}$$

This implications are, in general, strict. However, for $c_0$ the last one is an equivalence, as we show next.

The following result was originally proved using the classical Bessaga-Pelczyński’s selection theorem and it was the motivation for Theorem 4.10.

Theorem 5.14

Let ${\textbf {x}}=(x_n)_n$ be a sequence in $c_0$. Then ${\mathcal {R}}\le _K {{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}} )$ iff $(x_n)_n$ is weakly null.

Proof

If ${\mathcal {R}}\le _K {{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}} )$, then ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})$ is tall and thus ${\textbf {x}}=(x_n)_n$ is weakly null by Proposition 5.13. Conversely, let $(x_n)_n$ be a weakly null sequence in $c_0$. By Lemma 5.7, we can assume that $x_n(k)\ge 0$ for all n and k. The corresponding measures are given by $\mu _k(\{n\})=x_n(k)$. Since each $x_n\in c_0$, $(\mu _k)_k$ is of type $c_0$ (see definition 4.9). Notice that $\varphi _{\textbf {x}}=\sup _k\mu _k$ and thus ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})={{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$. Condition (iii) in Theorem 4.10 (namely, $\lim _n\mu _k(\{n\})=0$ for all k) is the translation of being weakly null in $c_0$. Thus ${\mathcal {R}}\le _K{{\,\mathrm{\mathsf FIN}\,}}(\varphi _{\textbf {x}})$. $\square$

The previous result naturally suggests the following.

Question 5.15

Which Banach spaces satisfies that ${\mathcal {R}}\le _K {{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}} )$ for every weakly null sequence ${\textbf {x}}=(x_n)_n$?

In relation to the previous question. Let ${\textbf {x}}=(x_n)_n$ be the usual unit basis of $l_2$ (which is weakly null). Since $l_2$ does not contain copies of $c_0$, ${{\,\mathrm{{\mathcal {B}}}\,}}({\textbf {x}})={{\,\mathrm{\mathsf FIN}\,}}$. Moreover, the same happens in $l_\infty$, as this space contains isomorphic copies of every separable Banach space.