Lower Density Type Operators with Borel Values

Abstract

We prove that the lower density operator associated with intensity points on the real line has Borel values. We also prove that the simple density operator and the complete density operator have Borel values.

1 Preliminaries

Assume that $\Sigma$ is a $\sigma$-algebra of subsets of a set $X\ne \emptyset$ and let $\mathcal J\subsetneq \Sigma$ be a $\sigma$-ideal. We denote $A\sim B$ when the symmetric difference $A\bigtriangleup B$ belongs to $\mathcal J$. Note that $\sim$ is an equivalence relation on $\Sigma$. We say that a mapping $\Phi :\Sigma \rightarrow \Sigma$ is a lower density operator with respect to $\mathcal J$ if the following conditions are satisfied:

• (I) $\Phi (X)=X$ and $\Phi (\emptyset )=\emptyset$,

• (II) $A\sim B\implies \Phi (A)=\Phi (B)$ for every $A,B\in \Sigma$,

• (III) $A\sim \Phi (A)$ for every $A\in \Sigma$,

• (IV) $\Phi (A\cap B)=\Phi (A)\cap \Phi (B)$ for every $A,B\in \Sigma$.

A standard example of a lower density operator deals with the $\sigma$-algebra ${{\mathcal {L}}}$ of Lebesgue measurable subsets of ${\mathbb {R}}$ and the $\sigma$-ideal ${{\mathcal {N}}}$ of null sets; see [8]. The respective operator $\Phi :{\mathcal L}\rightarrow {{\mathcal {L}}}$ assigns to $A\in {\mathcal {L}}$ its set of density points. It is known (cf. [13]) that the value $\Phi (A)$ is always a Borel set of type $F_{\sigma \delta }$. The exact version of this example can also be considered for the Cantor space $\{0,1\}^{\mathbb {N}}$ with the respective product measure (see [1] and [6, Exercise 17.9]).

New important examples of lower density operators have been obtained thanks to a reformulation of the classical notion of a Lebesgue density point proposed in [12] and [9]. Before we describe this idea, we recall the notion of convergence with respect to a $\sigma$-ideal, introduced in [11]. Given a pair $\Sigma , \mathcal J$ as above, we say that a sequence $(f_n)$ of $\Sigma$-measurable functions $f_n:X\rightarrow {\mathbb {R}}$, $n\in {\mathbb {N}}$, converges to a $\Sigma$-measurable function $f:X\rightarrow {\mathbb {R}}$ with respect to $\mathcal J$ whenever every subsequence $(f_{n_k})$ of $(f_n)$ contains a subsequence $(f_{n_{k_p}})$ such that

$$\begin{aligned} \{x:f_{n_{k_p}}(x)\nrightarrow f(x)\}\in \mathcal J, \end{aligned}$$

that is $f_{n_{k_p}}(x)\rightarrow f(x)$ holds $\mathcal J$-almost everywhere.

Note that, if $X:=(-1,1)$, $\Sigma :={\mathcal L}\upharpoonright X$ and ${\mathcal J}:={\mathcal N}\upharpoonright X$, this convergence is equivalent to convergence in measure, by the Riesz theorem. We use the fact that Lebesgue measure is finite on $(-1,1)$. Here we put ${\mathcal L}\upharpoonright X:=\{ A\in {{\mathcal {L}}}:A\subseteq X\}$ and ${{\mathcal {N}}}\upharpoonright X:=\{ A\in {{\mathcal {N}}}:A\subseteq X\}$.

For $A\subseteq {\mathbb {R}}$ and $\alpha , x\in {\mathbb {R}}$ we denote $\alpha A:=\{\alpha t:t\in A\}$ and $A\pm x:=\{t\pm x:t\in A\}$. The characteristic function of A will be written as $\chi _A$. By $\lambda$ we denote Lebesgue measure on ${\mathbb {R}}$. Let $x\in {\mathbb {R}}$ and $A\in \mathcal L$. From the original definition stating that x is a density point of A one can easily deduce the following equivalent version:

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\lambda \left( A\cap \left( x-\frac{1}{n},x +\frac{1}{n}\right) \right) }{\frac{2}{n}}=1 \end{aligned}$$

which in turn is equivalent to each of the next two conditions:

$$\begin{aligned} \lim _{n\rightarrow \infty }\lambda (n(A-x)\cap (-1,1))=2; \end{aligned}$$

$$\begin{aligned} \text{ the } \text{ sequence } \left( \chi _{n(A-x)\cap (-1,1)}\right) _{n\in {\mathbb {N}}} \text{ converges } \text{ in } \text{ measure } \text{ to } \chi _{(-1,1)}. \end{aligned}$$

(1)

This last condition can be applied to the case of the Baire category where A is replaced by a set with the Baire property and convergence in measure is replaced by convergence with respect to the $\sigma$-ideal of meager sets in ${\mathbb {R}}$. This idea turned our fruitful since it initiated broad investigations of density points and density like topologies in the Baire category sense (see [3, 9, 13]).

Let us describe some datails now. By $\mathcal B$ we denote the $\sigma$-algebra of subsets of ${\mathbb {R}}$ with the Baire property, and by $\mathcal M$ – the $\sigma$-ideal of meager subsets of ${\mathbb {R}}$. A set $A\in \mathcal B$ can be written as the symmetric difference $U\bigtriangleup E$ where U is open and $E\in \mathcal M$. Moreover, U can be chosen regular open and then the above expression is unique; see [8]. The unique regular open set U in the expression $A=U\bigtriangleup E$ will be denoted by $\widetilde{A}$ and called the regular open part of A. If $\Sigma :=\mathcal B$ and $\mathcal J:=\mathcal M$, then convergence with respect to the $\sigma$-ideal $\mathcal M$ will be called convergence in (the Baire) category.

We say that 0 is an $\mathcal M$-density point of a set $A\in \mathcal B$ whenever the sequence of characteristic functions $(\chi _{nA\cap (-1,1)})$ converges in category to the constant function equal to 1 on $(-1,1)$. If, in this statement, the constant function equal to 1 is replaced by the constant function equal to 0, we say that number 0 is an $\mathcal M$-dispersion point of A. In one-sided versions of these definitions one should replace $(-1,1)$ by $(-1,0)$ and (0, 1). Given $x\in {\mathbb {R}}$, we say that x is an $\mathcal M$-density (respectively, $\mathcal M$-dispersion) point of $A\in \mathcal B$ whenever 0 is an $\mathcal M$-density (respectively, $\mathcal M$-dispersion) point of $A-z$. Note that if, in the definition of $\mathcal M$-density point, we replace the pair $\mathcal B,\mathcal M$ by $\mathcal L,\mathcal N$, we obtain the usual notion of density point. Observe that x is an $\mathcal M$-density point of A if and only if x is an $\mathcal M$-dispersion point of the complementary set $A^c$, and in the last case, $A^c$ can be replaced by its regular open part $\widetilde{A^c}$.

Operator $\Phi _{\mathcal M}$ assigning to each $A\in \mathcal B$ its set of $\mathcal M$-density points is a lower density operator [9]. A nontrivial question whether its values are Borel sets was settled positively in [2] where a characterization by Łazarow [7] served as a basic tool. Another result of [2] states that the operator $\Phi _{\mathcal M}$ can be considered in the Cantor space $\{0,1\}^{\mathbb {N}}$ and it has analogous properties.

Note that, analogously as in the measure case, the family $\{A\in \mathcal B:A\subseteq \Phi _{\mathcal M}(A)\}$ forms a topology, called the $\mathcal I$-density topology (with $\mathcal I=\mathcal M$) which has been widely investigated by several authors (see [3, 9]).

The aim of our paper is to show some new instances of lower density or lower density like operators having Borel values. This makes a continuation of studies in [2]. In Sect. 2 we consider another Baire category notion of density point proposed in [14] and called an intensity point. It is essentially different from the notion of $\mathcal M$-density point [14, Thm 9]. It generates in a standard way a topology that is not homeomorphic to $\mathcal I$-density topology; see [14, Thm 29]. However, if one reformulates the definition of an intensity density point for the pair $\mathcal L,\mathcal N$, this would give the usual notion of a Lebesgue density point [14, Thm 3]. Our main result of Sect. 2 states that that the intensity operator has Borel values of type $F_{\sigma \delta \sigma \delta }$ which solves the first part of Problem 3 in [2].

In Sect. 3 we consider almost lower density operators $\Phi _s, \Phi _c:\mathcal L\rightarrow \mathcal L$ generated by simple and complete density points introduced in [15] and [16], respectively. We show that $\Phi _s$ and $\Phi _c$ have Borel values of type $G_{\delta \sigma \delta }$ and $G_{\delta \sigma }$, respectively.

2 The Intensity Operator has Borel Values

Recall the notion of an intensity point [14]. Suppose that $A\in \mathcal B$. For $n\in {\mathbb {N}}$ let

$$\begin{aligned} A_n:=\left( n(n+1)\left( A-\frac{1}{n+1}\right) \right) \cap (0,1). \end{aligned}$$

Define $f_n:(0,1)\rightarrow {\mathbb {R}}$ by $f_n:=\frac{1}{n}\sum _{i=1}^n\chi _{A_i}$ for $n\in {\mathbb {N}}$. We say that 0 is a right-hand intensity point of A if the sequence $(f_n)$ converges in category to the constant function equal to 1 on $(-1,1)$. We say that 0 is a rarefaction point of A if the sequence $(f_n)$ converges to 0 in category. Clearly, 0 is a right-hand intensity point of A if and only if it is a rarefaction point of the complement $A^c$ of A. The definition of a left-hand intensity and rarefaction points is self explaining (then we consider $(-1,0)$ instead of (0, 1)). We say that 0 is an intensity (rarefaction) point of A if it is both right-hand and left-hand intensity (rarefaction) point of A. At last we say that $x\in {\mathbb {R}}$ is an intensity (rarefaction) point of A whenever 0 is an intensity (rarefaction) point of $A-x$.

Let $\Phi _i$ assign to every set $A\in \mathcal B$ its set of intensity points. It is proved in [14, Thm 6] that $\Phi _i$ is a lower density operator from ${\mathcal {B}}$ into itself. We will call it briefly the intensity operator. Observe that x is an intensity point of A if and only if it is a rarefaction point of $\widetilde{A^c}$ (the regular open part of $A^c$).

We shall prove that the intensity operator is Borel-valued. We need a characterization of an intensity point which would play a similar role to that played by the result of E. Łazarow in the case of an $\mathcal M$-density point [7]. We shall deal with right-hand intensity and rarefaction points since the left-hand case is analogous.

Theorem 1

Let $(f_n)_{n\in {\mathbb {N}}}$ be a sequence of non-negative real-valued functions defined on (0, 1), having the Baire property. For $h>0$ and $n\in {\mathbb {N}}$, we put

$$\begin{aligned} f_n^h(x):=\left\{ \begin{array}{ll} h &{} \hbox { if}\,\,\, f_n(x)>h\\ 0 &{} \hbox { if}\,\,\, 0\le f_n(x)\le h. \end{array}\right. \end{aligned}$$

Denote $f_n^h$ by $f_n^{(k)}$ if $h=1/k$, $k\in {\mathbb {N}}$. Then the following conditions are equivalent:

(i):

$(f_n)_{n\in {\mathbb {N}}}$ converges to 0 in category;

(ii):

$(f_n^h)_{n\in {\mathbb {N}}}$ converges to 0 in category for each $h>0$;

(iii):

$(f_n^{(k)})_{n\in {\mathbb {N}}}$ converges to 0 in category for each $k\in {\mathbb {N}}$.

Proof

(i) $\Rightarrow$ (ii) If $(f_n)_{n\in {\mathbb {N}}}$ coverges to 0 in category, then $(f_n^h)_{n\in {\mathbb {N}}}$ also converges to 0 in category because $0\le f_n^h\le f_n$ for all $n\in {\mathbb {N}}$ and $h>0$.

Implication (ii) $\Rightarrow$ (iii) is obvious.

(iii) $\Rightarrow$ (i) From the assumption it follows that for each $k\in {\mathbb {N}}$ and for each increasing sequence $(n_m)_{m\in {\mathbb {N}}}$ of positive integers there exists a subsequence $(n_{m_p})_{p\in {\mathbb {N}}}$ for which

$$\begin{aligned} \left\{ x\in (0,1):\limsup _{p\rightarrow \infty }f^{(k)}_{n_{m_p}}(x)\ge h\right\} \in \mathcal M. \end{aligned}$$

We will use the denotation $(n_m^{(1)}), (n_m^{(2)}),\dots$ for successive subsequences in the inductive construction described below.

For $k=1$ pick a subsequence $(n_m^{(1)})$ such that

$$\begin{aligned} \left\{ x\in (0,1):\limsup _{m\rightarrow \infty }f^{(1)}_{n^{(1)}_m}(x)\ge 1\right\} \in \mathcal M. \end{aligned}$$

If we have constructed subsequences $(n_m^{(1)}), (n_m^{(2)}),\dots ,(n_m^{(k)})$, we pick a subsequence $(n_m^{(k+1)})$ of $(n_m^{(k)})$, for which

$$\begin{aligned} \left\{ x\in (0,1):\limsup _{m\rightarrow \infty }f^{(k+1)}_{n^{(k+1)}_m}(x)\ge \frac{1}{k+1}\right\} \in \mathcal M. \end{aligned}$$

Now, consider the diagonal sequence $(n_{m_p})_{p\in {\mathbb {N}}}$ for $(n_m^{(1)}),(n_m^{(2)}),\dots$. Then

$$\begin{aligned} \left\{ x\in (0,1):\limsup _{p\rightarrow \infty }f_{n_{m_p}}(x)>0\right\} \subseteq \bigcup _{k\in {\mathbb {N}}} \left\{ x\in (0,1):\limsup _{m\rightarrow \infty } f^{(k)}_{n^{(k)}_m}(x)\ge \frac{1}{k}\right\} . \end{aligned}$$

(2)

Indeed, let $x\in (0,1)$ and $\limsup _{p}f_{n_{m_p}}(x)>0$. Then there exists $h>0$ such that $f_{n_{m_p}}(x)>h$ for infinitely many p’s. If $k\in {\mathbb {N}}$ is such that $1/k<h$, then $f^{(k)}_{n^{(k)}_m}(x)\ge \frac{1}{k}$. Finally from (2) it follows that

$$\begin{aligned} \left\{ x\in (0,1):\limsup _{p\rightarrow \infty }f_{n_{m_p}}(x)>0\right\} \in \mathcal M. \end{aligned}$$

$\square$

We enumerate a fixed basis of the natural topology in (0, 1) as $((a_k, b_k))_{k\in {\mathbb {N}}}$. Then we enumerate a fixed basis of the natural topology in $(a_k,b_k)$ as $((c_{k,n}, d_{k,n}))_{n\in {\mathbb {N}}}$.

Theorem 2

Suppose that $(A_n)_{n\in {\mathbb {N}}}$ is a sequence of open subsets of (0, 1). Then the following conditions are equivalent:

(i):

the sequence $(\chi _{A_n})$ converges to 0 in category;

(ii):

for each $k\in {\mathbb {N}}$ there exist $\varepsilon >0$ and $n_0\in {\mathbb {N}}$ such that for each $n>n_0$ there exists $m\in {\mathbb {N}}$ with $d_{k,m}-c_{k,m}>\varepsilon$ and $A_n\cap (c_{k,m}-d_{k,m})=\emptyset$.

Proof

(i)$\Rightarrow$(ii) Suppose that (ii) does not hold. Then there exists $k_0\in {\mathbb {N}}$ such that for all $\varepsilon >0$ and $j\in {\mathbb {N}}$ there exists $n=n(\varepsilon ,j)>j$ such that for each $m\in {\mathbb {N}}$ if $d_{k_0,m}-c_{k_0,m}>\varepsilon$, then $A_{n(\varepsilon ,j)}\cap (c_{k_0,m}, d_{k_0,m})\ne \emptyset$. Hence for $\varepsilon =1$ and $j=1$ there exists $n_1=n(1,1)>1$, for $\varepsilon =1/2$ and $j=n_1$ there exists $n_2=n(1/2,n_1)>n_1$ and so on, which means that for each $p\in {\mathbb {N}}$, if $\varepsilon =1/p$ and $j=n_{p-1}$, then there exists $n_p=n(1/p,n_{p-1})$ such that, for each $m\in {\mathbb {N}}$, $d_{k_0,m}-c_{k_0,m}>1/p$ implies that $A_{n_p}\cap \left( c_{k_0,m}, d_{k_0,m}\right) \ne \emptyset$. Obviously, the sequence $(n_p)$ is increasing. Observe that, if $N_0\subseteq {\mathbb {N}}$ is an infinite set, then $\bigcup _{p\in N_0}A_{n_p}$ is open and dense in $(a_{k_0}, b_{k_0})$. Hence the sequence $(\chi _{A_n})$ does not converge to 0 in category.

(ii)$\Rightarrow$(i) To simplify notation we shall choose a subsequence, convergent $\mathcal M$-almost everywhere to 0, from the whole sequence $(\chi _{A_n})$. First, take $k=1$ and $(a_1,b_1)$. There exists $\varepsilon (1)>0$ and $n_0(1)$ such that for each $n>n_0(1)$ there exists $m_n\in {\mathbb {N}}$ with $(c_{1,m_n}, d_{1, m_n})\cap A_n=\emptyset$ and $d_{1,m_n}-c_{1,m_n}>\varepsilon (1)$. It is possible to choose an increasing sequence $S^{(1)}$ of positive integers for which there exists an interval $(u_1,v_1)\subseteq (a_1,b_1)$ such that $(u_1,v_1)\cap A_n=\emptyset$ for each n in $S^{(1)}$. Repeating this procedure for $k=2$, we find an interval $(u_2,v_2)\subseteq (a_2,b_2)$ and a subsequence $S^{(2)}$ of $S^{(1)}$ such that $(u_2,v_2)\cap A_n=\emptyset$ for each n in $S^{(2)}$. We construct inductively a sequence $(S^{(m)})_{m\in {\mathbb {N}}}$ of sequences where $S^{(m+1)}$ is a subsequence of $S^{(m)}$ for every m. Consider the diagonal sequence $(n_m)$ of $(S^{(m)})$. Then $(\chi _{A_{n_m}})$ converges to 0 on the dense open set $\bigcup _{k\in {\mathbb {N}}}(u_k,v_k)$, that is, it converges to 0, $\mathcal M$-almost everywhere. $\square$

Let us preserve our previous notation. Then we obtain the following corollary.

Corollary 3

Suppose that $(f_n)$ is a sequence of non-negative lower semicontinuous functions defined on (0, 1). Then for each $h>0$ the following conditions are equivalent:

(i):

the sequence $(f^h_n)_{n\in {\mathbb {N}}}$ converges to 0 in category;

(ii):

for each $k\in {\mathbb {N}}$ there are $p\in {\mathbb {N}}$ and $n_0\in {\mathbb {N}}$ such that for each $n>n_0$ there exists $m\in {\mathbb {N}}$ with $d_{k,m}-c_{k,m}>1/p$ and $\{x\in (0,1):f_n^h(x)>0\}\cap \left( c_{k,m},d_{k,m}\right) =\emptyset$.

Proof

Fix $h>0$. Since for $n\in {\mathbb {N}}$ the function $f_n$ is lower semicontinuous, the set

$$\begin{aligned} A_n:=\{x\in (0,1):f_n^h(x)>0\}=\left\{ x\in (0,1):\frac{1}{h}f_n^h(x)>0\right\} \end{aligned}$$

is open and $\frac{1}{h}f_n^h(x)=\chi _{A_n}$. Hence it is enough to apply Theorem 2. Additionally, we observe that a number $\varepsilon >0$ in Theorem 2(ii) can be chosen of the form 1/p for some $p\in {\mathbb {N}}$. $\square$

Theorem 4

For each set $A\in \mathcal B$, the set $\Phi _i(A)$ of all intensity points of A is a Borel set of type $F_{\sigma \delta \sigma \delta }$.

Proof

Fix $A\in \mathcal B$. Then $\Phi _i(A)= E^+\cap E^-$ where $E^+, E^-$ denote (respectively) the sets of right-hand and left-hand intensity points of A. We will show that $E^+$ is of type $F_{\sigma \delta \sigma \delta }$. The proof for $E^-$ is analogous, so it will be omitted.

For $n\in {\mathbb {N}}$ and $x\in (0,1)$ let $f_{n,x}(t):=\frac{1}{n}\sum _{i=1}^n\chi _{A_i(x)}(t)$ if $t\in (0,1)$ where

$$\begin{aligned} A_i(x):=\left( i(i+1)\left( \widetilde{A^c}-x-\frac{1}{i+1}\right) \right) \cap (0,1). \end{aligned}$$

Since sets $A_i(x)$ for $i\in {\mathbb {N}}$, are open, the function $f_{n,x}$ is lower semicontinuous.

From Theorem 1 it follows that $E^+$ consists of $x\in (0,1)$ such that $(f^{(r)}_{n,x})_{n\in {\mathbb {N}}}$ converges to 0 in category for every $r\in {\mathbb {N}}$ (one should replace $f^{(k)}_{n}$ by $f^{(k)}_{n,x}$ in condition (ii) of Theorem 1). Then observe that the statement “$(f^{(r)}_{n,x})_{n\in {\mathbb {N}}}$ converges to 0 in category” can be formulated in an equivalent form by the use of Corollary 3(ii). Consider the sequence $((c_{k,m}, d_{k,m}))_{m\in {\mathbb {N}}}$ introduced prior to Theorem 2. Given $p\in {\mathbb {N}}$, we enumerate as $((c^{p}_{k,m}, d^{p}_{k,m}))_{m\in {\mathbb {N}}}$ all intervals in $((c_{k,m}, d_{k,m}))_{m\in {\mathbb {N}}}$ for which $d_{k,m}-c_{k,m}>1/p$. Now, by the previous observation and Corollary 3(ii), we infer that x belongs to $E^+$ if and only if,

for all $r, k\in {\mathbb {N}}$ there exist $p, n_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \forall n>n_0\;\exists m\in {\mathbb {N}}\;\;\{t\in (0,1):f^{(r)}_{n,x}(t)>0\}\cap (c^p_{k,m},d^p_{k,m})=\emptyset . \end{aligned}$$

(3)

Our purpose is to show that the set

$$\begin{aligned} \{x\in (0,1):\{t\in (0,1):f^{(r)}_{n,x}(t)>0\}\cap (c^p_{k,m},d^p_{k,m})=\emptyset \} \end{aligned}$$

(4)

is closed since, by (3), this implies that $E^+$ is of type $F_{\sigma \delta \sigma \delta }$.

Claim 1. For a fixed $t\in (0,1)$, condition $f^{(r)}_{n,x}(t)>0$ is equivalent to $f^{(r)}_{n,x}(t)>1/r$. This in turn is equivalent to the following condition:

t

here exist an integer $j(r)\le n$, with $j(r)>\lfloor \frac{n}{r}\rfloor +1$, and a sequence $i_1<i_2<\dots <i_{j(r)}$ of integers in $\{1,\dots , n\}$ such that $t\in \bigcap _{j=1}^{j(r)}A_{i_j}(x)$.

Let us prove this last equivalence. Implication “$\Rightarrow$” follows from the fact that $f^{(r)}_{n,x}(t)$ is the arithmetic mean with the components equal to zeros or ones, so the number j(r) of ones should be such that $\frac{j(r)}{n}>\frac{1}{r}$, hence $j(r)>\lfloor \frac{n}{r}\rfloor +1$. The proof of the reverse implication is based on a similar reasoning.

From Claim 1 it follows that $\{t\in (0,1):f^{(r)}_{n,x}(t)>0\}$ is a finite union of sets of the form $\bigcap _{j=1}^{j(r)}A_{i_j}(x)$. The next claim is a simple observation.

Claim 2. The set $\{t\in (0,1):f^{(r)}_{n,x}(t)>0\}$ is disjoint from $(c^p_{k,m},d^p_{k,m})$ if and only if all components of the above-mentioned finite union are disjoint from $(c^p_{k,m},d^p_{k,m})$.

Now, observe that, to show the closedness of the set in (4), it suffices to prove that every set of the form $\{x\in (0,1):\bigcap _{j=1}^{j(r)}A_{i_j}(x)\cap (c^p_{k,m},d^p_{k,m})=\emptyset \}$ is closed. This is true since the complementary set

$$\begin{aligned} \left\{ x\in (0,1):\bigcap _{j=1}^{j(r)}\left( i_j(i_j+1)\left( \widetilde{A^c} -\frac{1}{i_j+1}\right) -i_j(i_j+1)x\right) \cap (c^p_{k,m},d^p_{k,m})\ne \emptyset \right\} \end{aligned}$$

is open. Indeed, the above formula says that the intersection of two open sets is nonempty. So, if $x_0$ belongs to the above set, then every x sufficiently close to $x_0$ is also in this set. $\square$

It would be interesting to establish whether the class $F_{\sigma \delta \sigma \delta }$ in Theorem 4 can be replaced by a lower class in the Borel hierarchy.

The definition of an intensity (rarefaction) point of a set $A\subseteq {\mathbb {R}}$ with the Baire property, introduced in [14], was motivated by the fact that, if we use a similar idea in the measure case, we obtain the classical notion of a density (dispersion) point. In fact in the measure case, the analogue for 0 to be a right-hand rarefaction point of a measurable set $A\subseteq {\mathbb {R}}$ means that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^n b_k=0 \end{aligned}$$

(5)

where $b_k:=k(k+1)\lambda \left( A\cap \left[ \frac{1}{k+1},\frac{1}{k}\right] \right)$ is the average density of A on $\left[ \frac{1}{k+1},\frac{1}{k}\right]$. By [14, Thm 3] we know that condition (5) is equivalent to the fact that 0 is the right-hand dispersion point of A.

Unfortunately, for the Cantor space $\{0,1\}^{\mathbb {N}}$ and the respective product probability measure, we are not able to find a reasonable analogue of condition (5). Therefore, we cannot find a counterpart of the notion of a rarefaction point of a set $A\subseteq \{0,1\}^{\mathbb {N}}$ with the Baire property. Also, the characterization of intensity points contained in the proof of Theorem 4 seems useless for this purpose. Hence the second part of Problem 3 in [2] remains open and we do not see a reasonable solution.

3 The Simple Density Operator and the Complete Density Operator have Borel Values

In [15] and [16], two versions of modified Lebesgue density points were introduced. A main idea was to replace convergence in measure stated in condition (1) by stronger kinds of convergence.

If we consider convergence almost everywhere in (1), we obtain the notion of a simple density point from [15]. So, a number $x\in {\mathbb {R}}$ is called a simple density point of a set $A\in \mathcal L$ if the sequence $\left( \chi _{n(A-x)\cap (-1,1)}\right) _{n\in {\mathbb {N}}}$ converges to $\chi _{(-1,1)}$ almost everywhere. Denote by $\Phi _s(A)$ the set of all simple density points of A. If $\Phi (A)$ is the set of usual (Lebesgue) density points of A, we have $\Phi _s(A)\subseteq \Phi (A)$ which is an easy consequence of the characterization (1) of $x\in \Phi (A)$.

From [15] we know that $\Phi _s:\mathcal L\rightarrow \mathcal L$ and $\Phi _s$ satisfies all axioms (I)–(IV) of a lower density operator except for (III) which is only partially fulfilled: we have $\Phi _s(A)\setminus A\in \mathcal N$ (which follows from $\Phi _s(A)\subseteq \Phi (A)$ and $\Phi (A){\setminus } A\in \mathcal N$) but $A\setminus \Phi _s(A)$ need not hold (see [15, Thm 1]). However, $\Phi _s$ is useful enough since the family $\tau _s:=\{A\in \mathcal L:A\subseteq \Phi _s(A)\}$ is a topology stronger than the natural topology and weaker than the usual density topology $\tau _d$ [15, Thm 2].

In general, a mapping $\Psi :\Sigma \rightarrow \Sigma$ fulfilling conditions (I), (II), (IV) and the condition $\Psi (A){\setminus } A\in \mathcal J$ (weaker than (III)) is called an almost lower density operator. See, for instance, [4, 10].

For a set $C\subseteq {\mathbb {R}}^2$ and $x\in {\mathbb {R}}$ we define the x-section of C by $C(x):=\{y:(x,y)\in C\}$.

Theorem 5

For each set $A\in \mathcal L$, the set $\Phi _s(A)$ of all simple density points of A is a Borel set of type $G_{\delta \sigma \delta }$.

Proof

We modify the proof of [15, Proposition 2]. Let $A\in \mathcal L$. We will check the Borelness of $\Phi _s(A)$. We may replace A by a $G_\delta$ set $A_0$ such that $A\sim A_0$ since then $\Phi _s(A)=\Phi _s(A_0)$.

Define $F:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$ by $F(x,y):=(x,y-x)$ and let $B:=F({\mathbb {R}}\times A_0)$. Since F is a homeomorphism, B is of type $G_\delta$. Note that $B=\{(x,y):y\in A_0-x\}$. For $n\in {\mathbb {N}}$, let

$$\begin{aligned} B_n=\{(x,ny):(x,y)\in B\}=\{(x,y):x\in {\mathbb {R}},\; y\in n(A_0-x)\}. \end{aligned}$$

Then $B_n$ is of type $G_\delta$. For $x\in {\mathbb {R}}$, condition $x\in \Phi _s(A_0)$ means that $\left( \chi _{n(A_0-x)\cap (-1,1)}\right) _{n\in {\mathbb {N}}}$ converges to $\chi _{(-1,1)}$ almost everywhere which is equivalent to $\lambda (D(x))=2$ where

$$\begin{aligned} D:=(\liminf _{n\in {\mathbb {N}}} B_n)\cap ({\mathbb {R}}\times (-1,1))=\left( \bigcup _{n\in {\mathbb {N}}}\bigcap _{k>n}B_k\right) \cap ({\mathbb {R}}\times (-1,1)). \end{aligned}$$

Then D is of type $G_{\delta \sigma }$. Observe that $\lambda (D(x))=2$ is equivalent to $\lambda (D(x))>2-\frac{1}{n}$ for every n. Finally, recall that, since D is of type $G_{\delta \sigma } =\Sigma ^0_3$ (in modern notation), the set

$$\begin{aligned} D_n:=\left\{ x:\lambda (D(x))>2-\frac{1}{n}\right\} \end{aligned}$$

is of the same Borel class (see [6, 17.25, 22.25]) for each $n\in {\mathbb {N}}$. Hence $\Phi _s(A_0) =\bigcap _{n\in {\mathbb {N}}} D_n$ is of type $G_{\delta \sigma \delta }$ as desired. $\square$

In [16], another almost lower density operator $\Phi _c:\mathcal L\rightarrow \mathcal L$ was introduced. This time, in condition (1), complete convergence was considered instead of convergence in measure. We say (see [5]) that a sequence $(f_n)$ of real-valued measurable functions converges completely to a measurable function f on $(-1,1)$ whenever, for every $\varepsilon >0$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{j=n}^\infty \lambda (\{y\in (-1,1):\vert f_j (y)-f(y)\vert >\varepsilon \})=0. \end{aligned}$$

Note that this condition implies that $(f_n)$ converges to f almost everywhere. Also, it can be written in an equivalent form as follows

$$\begin{aligned} \sum _{j=1}^\infty \lambda (\{y\in (-1,1):\vert f_j (y)-f(y)\vert >\varepsilon \})<\infty . \end{aligned}$$

Now, a number $x\in {\mathbb {R}}$ is called a complete density point of a set $A\in \mathcal L$ if the sequence $\left( \chi _{n(A-x)\cap (-1,1)}\right) _{n\in {\mathbb {N}}}$ converges completely to $\chi _{(-1,1)}$. Observe that this is equivalent to the following condition

$$\begin{aligned} \sum _{j=1}^\infty \lambda (j(A^c-x)\cap (-1,1))<\infty . \end{aligned}$$

(6)

Denote by $\Phi _c(A)$ the set of all complete density points of A. From [16] we know that $\Phi _c:\mathcal L\rightarrow \mathcal L$ and $\Phi _c$ is an almost lower density operator such that $\Phi _c(A)\subseteq \Phi _s(A)$ for all $A\in \mathcal L$ (moreover, this inclusion can be proper). Also, the family $\tau _c:=\{A\in \mathcal L:A\subseteq \Phi _c(A)\}$ is a topology stronger than the natural topology and weaker than the topology $\tau _s$ described above.

The following lemma seems to be known.

Lemma 6

For a measurable set $B\subset {\mathbb {R}}$, define $f_B(x):=\lambda ((B+x)\cap (-1,1))$ where $x\in {\mathbb {R}}$. Then the function $f_B$ is continuous.

Proof

First we show that $f_B$ is continuous at 0. Let $\vert x\vert <1$ and assume, for instance, that $x>0$. Clearly,

$$\begin{aligned} \lambda ((B+x)\cap (-1,1))= \lambda ((B\cap (-1-x,1-x)) \end{aligned}$$

and this value can be expressed as the sum

$$\begin{aligned} \lambda (B\cap (-1,1))-\lambda ((B\cap [1-x,1))+\lambda ((B\cap (-1-x,-1]). \end{aligned}$$

Hence $\vert f_B(x)-f_B(0)\vert =\vert -\lambda ((B\cap [1-x,1))+\lambda ((B\cap (-1-x,-1])\vert \le x$. The case $x<0$ is similar. Thus $\vert f_B(x)-f_B(0)\vert \le \vert x\vert$ which yields the desired continuity. Now, let $x_0\ne 0$. Observe that $f_B(x_0)=f_{B+x_0}(0)$, so the continuity of $f_B$ at $x_0$ follows from the previous part of our proof. $\square$

Theorem 7

For each set $A\in \mathcal L$, the set $\Phi _c(A)$ of all complete density points of A is a Borel set of type $F_{\sigma }$.

Proof

Let $A\in \mathcal L$. Then, by (6), a point $x\in {\mathbb {R}}$ is in $\Phi _c(A)$ if and only if

$$\begin{aligned} \exists k\in {\mathbb {N}}\;\; \forall n\in {\mathbb {N}}\;\;\,\sum _{j=1}^n \lambda (j(A^c-x)\cap (-1,1)) \le k. \end{aligned}$$

Hence

$$\begin{aligned} \Phi _c(A)=\bigcup _{k\in {\mathbb {N}}}\bigcap _{n\in {\mathbb {N}}}\left\{ x\in {\mathbb {R}}:\sum _{j=1}^n \lambda (j(A^c-x)\cap (-1,1)) \le k\right\} . \end{aligned}$$

(7)

It follows from Lemma 6 that every function $x\mapsto \lambda (j(A^c-x)\cap (-1,1))$ (for $j\in \{1,\dots , n\}$) is continuous, hence so is the function

$$\begin{aligned} x\mapsto \sum _{j=1}^n \lambda (j(A^c-x)\cap (-1,1)). \end{aligned}$$

Consequently, by (7), the set $\Phi _c(A)$ is of type $F_{\sigma }$. $\square$

We can ask whether evaluation of a Borel class in Theorems 5 and 7 can be improved. If we expect the negative answer for the case of $\Phi _c$, we should find a set $A\in \mathcal L$ such that $\Phi _c(A)$ is of class $F_{\sigma }{\setminus } G_{\delta }$.