Random Walk in Balls and an Extension of the Banach Integral in Abstract Spaces

Abstract

We describe the construction of a random walk in a Banach space $\mathbb {B}$ with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in $L^{p}\left( \mathbb {B}\right) ,\,1\le p\,<\infty ,$ to a random element $\xi ,$ which generates a probability measure with support contained in the unit ball $B\subset \mathbb {B}$. Moreover, we define the Banach integral with respect to the distribution of $\xi$ for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces $\ell ^{p},\,1\le p < \infty$ and $L^{p}[0,1],\,1< p < \infty$ are given.

1 Introduction

Let $\left\{ e_{i},i\ge 1\right\}$ be a complete orthonormal system (CONS) in a separable Hilbert space $\left( \mathbb {H},\left| \cdot \right| \right)$ and let $\pi _{n}\left( x\right) =\sum _{i=1}^{n}x_{i}e_{i}$ for $x=\sum _{i=1}^{\infty }x_{i}e_{i}\in \mathbb {H}$. Denote by $K_{n}$ and B the unit balls in $\mathbb {R}^{n}$ and $\mathbb {H}$, respectively.

In a paper published as addendum to the Saks monograph Theory of the Integral, Banach [1] described the most general form of a nonnegative linear functional F (satisfying additional conditions analogous to some properties of the Lebesgue integral, thus called by Banach $\mathfrak {L}$-integral), defined on the linear set $\mathcal {L}$ of bounded, Borel measurable functions $\varPhi {:}\,B\rightarrow \mathbb {R}$, namely

$$\begin{aligned} F\left( \varPhi \right) =\lim _{n\rightarrow \infty }F_{n}\left( \varPhi \right) , \end{aligned}$$

where

$$\begin{aligned}&F_{n}\left( \varPhi \right) =\int _{K_{n}}\varPhi \left( \pi _{n}\left( x\right) \right) g_{n}\left( x_{1},\ldots ,x_{n}\right) \mathrm{d}x_{1}\ldots \mathrm{d}x_{n},\\&g_{n}\left( x_{1},\ldots ,x_{n}\right) \\&\quad =\mathbb {1}_{K_{n}}\left( x_{1},\ldots ,x_{n}\right) \frac{g\left( x_{1}\right) g\left( x_{2}{/}\sqrt{1-x_{1}^{2}}\right) \ldots \, g\left( x_{n}{/}\sqrt{ 1-\left( x_{1}^{2}+\cdots +x_{n-1}^{2}\right) }\right) }{\sqrt{\left( 1-x_{1}^{2}\right) \cdot \left[ 1-\left( x_{1}^{2}+x_{2}^{2}\right) \right] \cdot \ldots \cdot \left[ 1-\left( x_{1}^{2}+\cdots +x_{n-1}^{2}\right) \right] }}, \end{aligned}$$

$g{:}\,\left[ -1,1\right] \rightarrow \left[ 0,\infty \right)$ is Borel measurable and integrable with $\int _{-1}^{1}g\left( t\right) \mathrm{d}t=1,$ and $\mathbb {1}_{A}$ denotes the indicator of the set A.

In fact, Banach [1] considered only the case when g is the density of the uniform distribution on $[-\,1,1]$, and a more general case was treated by Banek [2]. Furthermore, Banek [2] observed that

$$\begin{aligned} F_{n}\left( \varPhi \right) =E\varPhi \left( Z_{n}\right) , \end{aligned}$$

where $\left\{ Z_{n} \right\}$ is the so-called Banach random walk (BRW) in $B\subset \mathbb {H}$ given by the random linear combination $Z_{n}=\sum _{i=1}^{n}X_{i}e_{i},~ n\ge 1,$ of elements of CONS $\{e_{i} \}$ in $\mathbb {H}$, with coefficients $X_{i}$ that are dependent r.v.’s defined recursively as follows: $X_{1}$ is a r.v. having density g concentrated on the interval $\left[ -\,1,1\right]$, and if the r.v.’s $X_{1},\ldots ,X_{n-1}$ are already defined, then $X_{n}$ is defined as a r.v. with probability density $g\left( x_{n}{/}\sqrt{1-(X_{1}^{2}+\cdots +X_{n-1}^{2})}\right)$ in the random interval

$$\begin{aligned} \left[ -\sqrt{ 1-\left( X_{1}^{2}+\cdots +X_{n-1}^{2}\right) },\sqrt{1-\left( X_{1}^{2}+\cdots +X_{n-1}^{2}\right) }\right] . \end{aligned}$$

The last observation forms a probabilistic background to the purely deterministic Banach construction of the $\mathfrak {L}$-integral for a class of bounded, Borel measurable functions defined in $B\subset \mathbb {H}$.

It is worth mentioning that Banach [1] considered two various special cases: (1) the mapping $\varPhi$ is defined on a compact metric space and (2) $\varPhi$ is defined in the unit ball of a separable Hilbert space.

In this paper, we describe a generalized BRW $\left\{ Z_{n},n\ge 1\right\}$ with values in the unit ball of a Banach space. Moreover, we give a criterion for the existence of the Banach–Lebesgue integral

$$\begin{aligned} F\left( \varPhi \right) =\lim _{n\rightarrow \infty }F_{n}\left( \varPhi \right) \end{aligned}$$

(1)

in terms of the constructed BRW $\left\{ Z_{n},n\ge 1\right\}$, where $\varPhi$ is a bounded, Borel measurable real-valued function defined in the unit ball of a Banach space.

2 Banach Random Walk in a Banach space

Let $\left( \mathbb {B},\left\| \cdot \right\| \right)$ be an infinite-dimensional Banach space with a Schauder basis $\left\{ b_{n},n\ge 1\right\}$. Then each vector $x\in \mathbb {B}$ possesses a unique series expansion $x=\sum _{k=1}^{\infty }x_{k}b_{k},$ and thus, for all $n\ge 1$, the projections $\pi _{n}{:}\,\mathbb {B}\rightarrow \mathbb {B},$ given by $\pi _{n}\left( x\right) =\sum _{k=1}^{n}x_{k}b_{k}$, are well defined. Denote

$$\begin{aligned} B_{n}=\left\{ \pi _{n}\left( x\right) \in \mathbb {B}{:}\,\left\| \pi _{n}\left( x\right) \right\| \le 1\right\} ,\quad B=\left\{ x\in \mathbb { B}{:}\,\left\| x\right\| \le 1\right\} , \end{aligned}$$

and put

$$\begin{aligned} \alpha _{1}=\inf \left\{ t\in \mathbb {R}{:}\,\left\| tb_{1}\right\| \le 1\right\} ,\quad \beta _{1}=\sup \left\{ t\in \mathbb {R}{:}\,\left\| tb_{1}\right\| \le 1\right\} \left( =-\alpha _{1}\right) . \end{aligned}$$

Furthermore, given any point $\pi _{n-1}\left( x\right) \in B_{n-1},\,n\ge 2$, define inductively

$$\begin{aligned} \alpha _{n}= & {} \alpha _{n}(\pi _{n-1}\left( x\right) ) =\, \inf \left\{ t\in \mathbb {R}{:}\,\left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1\right\} ,\\ \beta _{n}= & {} \beta _{n}(\pi _{n-1}\left( x\right) ) =\, \sup \left\{ t\in \mathbb {R}{:}\,\left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1\right\} . \end{aligned}$$

It is clear that $\alpha _{n}\le 0\le \beta _{n}$, and $\left[ \alpha _{n},\beta _{n}\right] ,\,n\ge 1$, are bounded intervals in $\mathbb {R}$, for

$$\begin{aligned} \left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1 \,\, \Rightarrow \,\, \left| t\right| \cdot \left\| b_{n}\right\| \le \left\| \pi _{n-1}\left( x\right) \right\| +1, ~ n\ge 1, \end{aligned}$$

where $\pi _{n-1}\left( x\right) = \pi _{0}\left( x\right) = 0$ for $n=1$. Obviously, $\alpha _{n}$ and $\beta _{n}$ depend on $\pi _{n-1}\left( x\right) \in B_{n-1}$ and $b_n$, and in general the intervals $\left[ \alpha _{n},\beta _{n}\right] ,\,n\ge 2$, need not be symmetric about 0. In addition, it may happen that for some $n\ge 2$ the interval $\left[ \alpha _{n},\beta _{n}\right]$ reduces to the single point $\left[ 0,0\right] =\left\{ 0\right\}$. To fix a standard length of the first interval, without loss of generality we may and do assume that $\left\| b_{1}\right\| =1$ (but we do not require that $\left\| b_{n}\right\| =1$ for all $n\ge 2$). In such a situation, $\alpha _{1}=-1$ and $\beta _{1}=1$.

Let $G_{n},\,n\ge 1$, be arbitrary probability distributions concentrated on $\left[ -\,1,1\right] \subset \mathbb {R}$, i.e., $G_{n}\left( \left[ -\,1,1\right] \right) =1$ for all $n\ge 1$. Define inductively on a probability space $\left( \varOmega , \mathcal {F},P\right)$ a sequence of (dependent) real r.v.’s $\left\{ X_{n},n\ge 1\right\}$ and, associated with it, a sequence of $\mathbb {B}$-valued random elements (r.e.) $\left\{ Z_{n},n\ge 1\right\}$. Namely, let $X_{1}$ be a r.v. with distribution $G_{1}$ and let $Z_{1}=X_{1}b_{1}$. Then $X_{1}\left( \omega \right) \in \left[ \alpha _{1},\beta _{1}\right] =\left[ -1,1\right]$ a.s., and thus, we may define $X_{2}$ as a r.v. distributed according to $G_{2},$ scaled linearly in such a way that it is concentrated on $\left[ \alpha _{2},\beta _{2} \right] =\left[ \alpha _{2}\left( Z_{1}\left( \omega \right) \right) ,\beta _{2}\left( Z_{1}\left( \omega \right) \right) \right]$. In other words, $X_{2}$ is a r.v. with distribution function

$$\begin{aligned} G_{2}\left( \frac{2t- \left[ \beta _{2}\left( Z_{1}\left( \omega \right) \right) +\alpha _{2}\left( Z_{1}\left( \omega \right) \right) \right] }{\beta _{2}\left( Z_{1}\left( \omega \right) \right) -\alpha _{2}\left( Z_{1}\left( \omega \right) \right) } \right) ,~ t\in \mathbb {R}, \end{aligned}$$

whenever $\beta _2 - \alpha _2 > 0$, and then we put $Z_{2}=X_{1}b_{1}+X_{2}b_{2}$. Next, given any value $X_{2}\left( \omega \right)$, and a fortiori$Z_{2}\left( \omega \right) \in B_{2}$ a.s., we define $X_{3}$ as a r.v. with distribution $G_{3}$ scaled linearly in such a way that it is concentrated on the interval$\left[ \alpha _{3},\beta _{3}\right] =\left[ \alpha _{3}\left( Z_{2}\left( \omega \right) \right) ,\beta _{3}\left( Z_{2}\left( \omega \right) \right) \right] ,$ and then put $Z_{3}=X_{1}b_{1}+X_{2}b_{2}+X_{3}b_{3},$ etc. More generally, if $X_{1}, \ldots ,X_{n-1}$ and $Z_{1}, \ldots ,Z_{n-1}$ are already defined in such a manner that $Z_{n-1}\left( \omega \right) \in B_{n-1}$ a.s., then $X_{n}$ is a r.v. with distribution function

$$\begin{aligned} G_{n}\left( \frac{2t- \left[ \beta _{n}\left( Z_{n-1}\left( \omega \right) \right) +\alpha _{n}\left( Z_{n-1}\left( \omega \right) \right) \right] }{\beta _{n}\left( Z_{n-1}\left( \omega \right) \right) -\alpha _{n}\left( Z_{n-1}\left( \omega \right) \right) } \right) , ~ t\in \mathbb {R}, \end{aligned}$$

provided $\beta _n - \alpha _n > 0$ and $Z_{n}=X_{1}b_{1}+X_{2}b_{2}+\cdots +X_{n}b_{n} = Z_{n-1} + X_{n}b_{n}.$

As was already mentioned, it may happen that for some $n\ge 1$ and $Z_{n}\left( \omega \right) \in B_{n}$ the interval $\left[ \alpha _{n+1},\beta _{n+1}\right] = \left[ \alpha _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right]$ is reduced to the single point $\left\{ 0\right\}$; in this situation, we assume that the distribution $G_{n+1}$ is transformed in such a way that it assigns the unit mass to the one point set $\{0\}$. Although in such a case $Z_{n+1}(\omega ) = Z_n (\omega )$, the next random interval $[\alpha _{n+2},\beta _{n+2} ] = [\alpha _{n+2}\left( Z_{n+1}(\omega )\right) ,\beta _{n+2}\left( Z_{n+1}(\omega )\right) ]$, defined by means of the successive basic vector $b_{n+2}$, need not be equal to $\{0\}$; thus, the process is still continued.

Definition 1

The sequence of $\mathbb {B}$-valued r.e.’s $\left\{ Z_{n},n\ge 1\right\}$ obtained in the way described above is called Banach random walk (BRW) in a Banach space $\mathbb {B}$.

It seems that the main idea of Banach’s [1] construction of $\mathfrak {L}$-integral was the symmetry of mappings corresponding to the symmetry of Lebesgue measures in $\mathbb {R}^{n},\,n\ge 1$, which led to convergence of the integral functional in (1). Therefore, we introduce in addition the following notion:

Definition 2

The Schauder basis $\left\{ b_{n},n\ge 1\right\}$ is called quasi-orthogonal, if

$$\begin{aligned} \beta _{n+1}(\pi _{n}\left( x\right) )= -\alpha _{n+1}\left( \pi _{n}\left( x\right) \right) \end{aligned}$$

(2)

for all $n\ge 1$ and $x\in \mathbb {B}$ such that $\pi _{n}\left( x\right) \in B_n$.

Recall that the basis $\left\{ b_{n},n\ge 1\right\}$ in a Banach space $(\mathbb {B}, \Vert \cdot \Vert )$ is said to be unconditional, if for all $n\ge 1,\,x_{k}\in \mathbb {R}$ and $\epsilon _{k}=\pm 1,\,1\le k\le n,$ we have

$$\begin{aligned} \left\| \sum \limits _{k=1}^{n}\epsilon _{k}x_{k}b_{k}\right\| =\left\| \sum \limits _{k=1}^{n}x_{k}b_{k}\right\| . \end{aligned}$$

(3)

The definition of a quasi-orthogonal Schauder basis in a Banach space seems to be similar to the condition defining an unconditional basis, but in spite of this, these two notions are not equivalent. If the basis $\left\{ b_{n},n\ge 1\right\}$ is unconditional, i.e., (3) holds, then it is obviously quasi-orthogonal, but the converse need not be true. To explain the notion of quasi-orthogonality, below we present the construction of relevant examples of Banach spaces with quasi-orthogonal Schauder bases which are not unconditional. It should be pointed out that many familiar Banach spaces possess unconditional Schauder bases (thus in fact quasi-orthogonal) consisting of unit vectors, but the class of Banach spaces with quasi-orthogonal bases is substantially larger than the class of spaces with unconditional bases.

3 Banach Spaces with Quasi-Orthogonal Bases

The quasi-orthogonal basis is constructed sequentially, step by step: given any basic vectors $b_{1},b_{2},\ldots ,b_{n},$ the element $b_{n+1}$ of the basis is chosen in such a way that for arbitrary $x_{1},x_{2},\ldots ,x_{n} \in \mathbb {R}$ satisfying condition $x_{1}b_{1}+ \cdots +x_{n}b_{n}=\pi _n (x)\in B_n,$ Eq. (2) is satisfied. As will be seen condition (3) for this property is not necessary.

1. Spaces of bounded sums and conditionally convergent series

Let $S=\mathbb {R}^{\mathbb {N}}=\left\{ x=\left( x_{1},x_{2},\ldots \right) {:}\,x_{k}\in \mathbb {R}\text { for all }k\ge 1\right\} \$be the set of all infinite sequences of real numbers. Define a function $\left\| \cdot \right\| {:}\,S\rightarrow \left[ 0,\infty \right]$ by

$$\begin{aligned} \left\| x\right\|= & {} \sup \left\{ \left| x_{1}+x_{2}\right| ,\left| x_{1}-x_{2}\right| ,\left| x_{1}+x_{2}+x_{3}\right| ,\left| x_{1}+x_{2}-x_{3}\right| ,\right. \\&\qquad \left. \ldots ,\left| x_{1}+\cdots +x_{n-1}+x_{n}\right| ,\left| x_{1}+\cdots +x_{n-1}-x_{n}\right| ,\ldots \right\} \\= & {} \sup \left\{ \left| x_{1}\right| + \left| x_{2}\right| , \left| x_{1}+x_{2}\right| + \left| x_{3}\right| , \ldots , \left| x_{1}+\cdots +x_{n}\right| + \left| x_{n+1} \right| , \ldots \right\} , \end{aligned}$$

and next put

$$\begin{aligned} S_{b}=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S{:}\,\left\| x\right\| <\infty \right\} . \end{aligned}$$

Then $\left\| \cdot \right\|$ is a norm in $S_{b}$, and $\left( S_{b},\left\| \cdot \right\| \right)$ is a Banach space. The space $S_{b}$ consists of all bounded sequences $\left( x_{1},x_{2},\ldots \right) \in S$ of real numbers with bounded partial sums $s_{n}=x_{1}+\cdots +x_{n};$ namely, if $x\in S_{b}$ and $\left\| x\right\| =M<\infty$, then $\left| x_{n}\right| \le M$ and $\left| s_{n}\right| \le M$ for all $n\ge 1$. Conversely, if there exists a constant $0\le M<\infty$ such that $\left| x_{n}\right| \le M$ and $\left| s_{n}\right| \le M$ for all $n\ge 1,$ then $\left| s_{n}\pm x_{n+1}\right| \le \left| s_{n}\right| +\left| x_{n+1}\right| \le 2M,$ and thus, $\left\| x\right\| \le 2M,$ i.e., $x=\left( x_{1},x_{2},\ldots \right) \in S_{b}.$ Therefore, $\left( S_{b},\left\| \cdot \right\| \right)$ may be called the space of bounded sums.

However, the space $S_{b}$ of sequences of real numbers is not separable. To show this, consider the family $2^{\mathbb {N}}$ of all the subsets of the set $\mathbb {N}=\left\{ 1,2,\ldots \right\} ,$ and for $\emptyset \ne A=\left\{ k_{1},k_{2},\ldots \right\} \subseteq \mathbb {N}$, where $k_1< k_2 < \cdots$, define $x_{A}$ as the sequence with terms $x_{k_{2j-1}}=1,\,x_{k_{2j}}=-1,\,j\ge 1,$ and $x_{i}=0$ otherwise, along with $x_{\emptyset }=(0,0,\ldots ).$ Then $\left\| x_{A}\right\| \le 2$, while $\left\| x_{A}-x_{B}\right\| \ge 1$ whenever $A,B\subseteq \mathbb {N},\,A\ne B.$ Since the set $2^{\mathbb {N}}$ is uncountable, the space $S_{b}$ is nonseparable. Hence in the context of our requirements, the space $S_{b}$ is inadequate.

Consider the set

$$\begin{aligned} S_{c}=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S_{b}{:}\,\text { the series } \sum \limits _{k}x_{k}\text { converges }\right\} . \end{aligned}$$

In other words, $S_{c}$ is the set of all sequences $x=\left( x_{1},x_{2},\ldots \right) \in S_{b}$ for which a finite limit $\lim _{n}s_{n}=s\in \mathbb {R}$ exists. It can be easily verified that $\left( S_{c},\left\| \cdot \right\| \right)$ is also a Banach space. Moreover, the space $S_{c}$ is separable. Indeed, the set of elements

$$\begin{aligned} \left\{ e_{n}=\left( \delta _{kn},k\ge 1\right) ,\,n=1,2,\ldots \right\} , \end{aligned}$$

where $\delta _{kn}=0$ for $k\ne n$ and $\delta _{nn}=1,$ is a basis of the space $S_{c},$ and finite linear combinations of vectors $e_{n}$ with rational coefficients from a countable dense subset in $S_{c}.$ Moreover, from the definition of the norm $\left\| \cdot \right\|$ it follows that the basis $\left\{ e_{n},\,n\ge 1\right\}$ is quasi-orthogonal, but it is not unconditional, because sums of the form $\epsilon _{1}x_{1}+\epsilon _{2}x_{2}+\cdots +\epsilon _{n}x_{n},\,x=\left( x_{1},x_{2},\ldots \right) \in S_{c},$ need not be convergent for all combinations of signs $\epsilon _{k}=\pm 1$. Taking into account the above properties, $\left( S_{c},\left\| \cdot \right\| \right)$ may be called the Banach space of conditionally convergent series.

Since the existence of $\lim _{n}s_{n}=s\in \mathbb {R}$ implies that $\lim _{n}x_{n}=0$, we conclude that $x\in S_{c}\Rightarrow x\in c_{0}.$ The Banach space $c_{0}$ of sequences of real numbers convergent to 0 is usually considered with the supremum norm $\left| x\right| _{\infty }=\sup \left\{ \left| x_{1}\right| ,\left| x_{2}\right| ,\ldots \right\} ,$ but the two norms $\left\| \cdot \right\|$ and $\left| \cdot \right| _{\infty }$ restricted to $S_{c}$ are not equivalent. To see this, consider the sequence of points $\left\{ x^{\left( n\right) },n\ge 1\right\} ,$

$$\begin{aligned} x^{\left( 1\right) }= & {} \left( 1,-1/2,1/3,-1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\ x^{\left( 2\right) }= & {} \left( 1,1/2,1/3,-1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\ x^{\left( 3\right) }= & {} \left( 1,1/2,1/3,1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\&\qquad \vdots \end{aligned}$$

and put $x^{\left( \infty \right) }=\left( 1,1/2,1/3,1/4,1/5,1/6,\ldots ,1/n,1/\left( n+1\right) ,\ldots \right) .$

Then $\left\| x^{\left( n\right) }\right\| <\infty$ for all $n\ge 1,$ so that $\left\{ x^{\left( n\right) },n\ge 1\right\} \subset S_{c}.$ Moreover, $\left| x^{\left( n\right) }-x^{\left( \infty \right) }\right| _{\infty }=1/n\rightarrow 0,$ while $\left\| x^{\left( n\right) }-x^{\left( \infty \right) }\right\| =\infty ,\,n=1,2,\ldots$ which is a consequence of the fact that $\sum \nolimits _{n}1/n=\infty .$ Therefore, the inclusion $S_{c}\subset c_{0}$ is valid only for sets, but it is not true for Banach spaces, $\left( S_{c},\left\| \cdot \right\| \right) \varsubsetneq \left( c_{0},\left| \cdot \right| _{\infty }\right) .$

A similar effect as in the case of the space $S_{c}$ can be obtained for every fixed system of signs $\epsilon =\left( \epsilon _{1},\epsilon _{2},\ldots \right) \in \left\{ -1,1\right\} ^{\mathbb {N}}$ and the norm given by

$$\begin{aligned} \left\| x\right\| _{\epsilon }=\left\| \left( \epsilon _{1}x_{1},\epsilon _{2}x_{2},\epsilon _{3}x_{3},\ldots \right) \right\| ~ \text {for} ~ x=(x_1,x_2, \ldots ). \end{aligned}$$

In this way, we obtain a Banach space $\left( S_{c,\epsilon },\left\| \cdot \right\| _{\epsilon }\right) ,$ where

$$\begin{aligned} S_{c,\epsilon }=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S{:}\,\left\| x\right\| _{\epsilon }<\infty \text { and the series }\sum \limits _{k} \epsilon _{k}x_{k}\text { converges}\right\} . \end{aligned}$$

The basis $\left\{ e_{n},\,n\ge 1\right\}$ in $S_{c,\epsilon }$ is quasi-orthogonal, but it is not unconditional. Note now that $\ell ^{1}\subseteq S_{c,\epsilon }$ for each $\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}$, therefore $\ell ^{1}\subseteq \bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{ \mathbb {N}}}S_{c,\epsilon }.$ On the other hand, assuming that $x=\left( x_{1},x_{2},\ldots \right) \in \displaystyle \bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }$ and taking $\epsilon ^{(x)}=\left( \mathrm {sign\,}x_{1}, \mathrm {sign\,}x_{2},\ldots ,\mathrm {sign\,}x_{n},\ldots \right)$ we have

$$\begin{aligned} \epsilon _{1}^{(x)}x_{1}+\epsilon _{2}^{(x)}x_{2}+\cdots +\epsilon _{n}^{(x)}x_{n}=\left| x_{1}\right| +\left| x_{2}\right| +\cdots +\left| x_{n}\right| , ~ n\ge 1. \end{aligned}$$

Hence, we infer that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left| x_{n}\right| =\sup \limits _{n}\left\{ \left| \epsilon _{1}^{(x)}x_{1}+\epsilon _{2}^{(x)}x_{2}+\cdots +\epsilon _{n}^{(x)}x_{n}\right| \right\} \le \left\| x\right\| _{\epsilon ^{(x)}}, \end{aligned}$$

where $\epsilon ^{(x)}\in \left\{ -1,1\right\} ^{\mathbb {N}}$, thus $x\in \ell ^{1}.$ Consequently, $\ell ^{1}=\bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }.$

Define next a function $\left\| \cdot \right\| _{\cap }{:}\,\bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }\rightarrow \left[ 0,\infty \right]$ by the formula: $\left\| x\right\| _{\cap }=\sup \left\{ \left\| x\right\| _{\epsilon },\epsilon \in \left\{ -1,1\right\} ^{ \mathbb {N}}\right\}$. Since

$$\begin{aligned} \left| \epsilon _{1}x_{1}+\epsilon _{2}x_{2}+\cdots +\epsilon _{n-1}x_{n-1}\pm \epsilon _{n}x_{n}\right| \le \left| x_{1}\right| +\left| x_{2}\right| +\cdots +\left| x_{n}\right| \end{aligned}$$

for each $\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}},$ where the inequality $\le$ may be replaced by the equality whenever $\epsilon _{k}=\mathrm {sign\,}x_{k},\,1\le k\le n-1$ and $\pm \,\epsilon _{n}=\mathrm {sign\,}x_{n}$, the function $\left\| \cdot \right\| _{\cap }$ assumes only finite values and in fact $\left\| x\right\| _{\cap }=\sum \nolimits _{n}\left| x_{n}\right| ,$ i.e., $\left\| \cdot \right\| _{\cap }$ is the norm equal precisely to the norm $\left| x\right| _{1}=\sum \nolimits _{n}\left| x_{n}\right|$ of the space $\ell ^{1}.$ Therefore, one can write

$$\begin{aligned} \left( \ell ^{1},\left| \cdot \right| _{1}\right) =\bigcap \limits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}}\left( S_{c,\epsilon },\left\| \cdot \right\| _{\epsilon }\right) . \end{aligned}$$

In this sense, the Banach space $\left( \ell ^{1},\left| \cdot \right| _{1}\right)$ in comparison with the space $\left( S_{c},\left\| \cdot \right\| \right)$ is “relatively small”. It is also worth mentioning that the basis $\left\{ e_{n},n\ge 1\right\}$ in $S_{c}$ (as well as in $S_{c,\epsilon }$) is monotone, i.e., for every choice of scalars $\{x_{n},n\ge 1\}\subseteq \mathbb {R}$, the sequence of numbers $\left\{ \Vert \sum \nolimits _{k=1}^{n}x_{n}e_{n}\Vert ,n\ge 1\right\}$ is nondecreasing.

Recall now that a basis $\{b_{n},n\ge 1\}$ of a Banach space $(\mathbb {B},\Vert \cdot \Vert )$ is called boundedly complete if, for every sequence of scalars $\{x_{n},n\ge 1\}\subseteq \mathbb {R}$ such that $\sup _{n\ge 1}\Vert \sum \nolimits _{k=1}^{n}x_{k}b_{k}\Vert <\infty$, the series $\sum \nolimits _{n=1}^{\infty }x_{n}b_{n}$ converges in norm of $\mathbb {B}$. Unfortunately, the basis $\{e_{n},n\ge 1\}$ in $S_{c}$ or $S_{c,\epsilon }$ is not boundedly complete.

2. Spaces of conditionally convergent series with rates of convergence

The space $S_{c}$ described here may be the prototype for a wide class of various spaces with quasi-orthogonal Schauder bases that are not unconditional. For instance, consider the spaces $S_{c}^{p}$ of (conditionally) summable sequences of real numbers spanned on vectors of the basis $\left\{ e_{n},n\ge 1\right\} ,$ with norms likewise in $\ell ^{p},\, 1\le p<\infty ,$ (that describe rates of convergence)

$$\begin{aligned} \left\| x\right\| _{p}=\left( \sum \limits _{~n=1}^{\infty }\left\| R_{n}\left( x\right) \right\| ^{p}\right) ^{1/p}, \end{aligned}$$

where $R_{n}\left( x\right) =\sum _{k=n}^{\infty }x_{k}e_{k}$ for $x=\left( x_{1},x_{2},\ldots \right) \in S$; spaces $S_{c,w}$ with norms determined by some positive weights $w=\left( w_{1},w_{2},\ldots \right) ,\,w_{i}>0,$

$$\begin{aligned} \left\| x\right\| _{1,w}=\sum \limits _{n=1}^{\infty }w_{n}\left\| R_{n}\left( x\right) \right\| , \end{aligned}$$

say geometrical weights $w=\left( w_{1},w_{2},\ldots \right) =\left( q^{1},q^{2},\ldots \right) ,\,q>0,$ or more generally, spaces $S_{c,w}^{p}$ equipped with norms

$$\begin{aligned} \left\| x\right\| _{p,w}=\left( \sum \limits _{~n=1}^{\infty }w_{n}\left\| R_{n}\left( x\right) \right\| ^{p}\right) ^{1/p}, ~ 1\le p<\infty , ~ \text {etc.} \end{aligned}$$

Since convergence of the series on the right-hand side of the definition of $\left\| x\right\| _{p}$ implies that partial sums of the series $\sum \nolimits _{n}x_{n}$ satisfy Cauchy’s criterion, the basis $\left\{ e_{n},n\ge 1\right\}$ in $\left( S_{c}^{p},\left\| \cdot \right\| _{p}\right)$ is evidently monotone and boundedly complete. Thus in view of Dunford’s theorem [4, Ch. III, §1, Th. 6, p. 64], these spaces possess the Radon–Nikodym property (RNP)—see [4, Ch. III, §1, p. 61] for the definition of this notion. Moreover, the equality

$$\begin{aligned} \left\| \sum \limits _{j=k}^{n}x_{j}e_{j}\right\| =\left\| \sum \limits _{j=k}^{n-1}x_{j}e_{j}-x_{n}e_{n}\right\| \end{aligned}$$

valid for all $\left( x_{1},x_{2},\ldots \right) \in S$ and $1\le k<n<\infty$ implies that the basis $\left\{ e_{n},n\ge 1\right\}$ in $\left( S_{c}^{p},\left\| \cdot \right\| _{p}\right)$ is again quasi-orthogonal, but it is not unconditional. A similar conclusion can be also derived for spaces equipped with norms $\left\| \cdot \right\| _{p,w},$ provided that w is a suitable sequence of weights.

3. Spaces of bounded sums of functions and conditionally convergent function series

Let $\left\{ q_{1}=0,q_{2}=1,q_{3},q_{4},\ldots \right\} \subset \left[ 0,1 \right] \subset \mathbb {R}$ be a countable set of numbers dense in [0, 1] (arranged in any order), and let $\left\{ e_{n}\left( t\right) ,n\ge 1\right\}$ be the system of Schauder hat functions in $\left[ 0,1\right]$ defined as follows: $e_{1}\left( t\right) =1-t$ and $e_{2}\left( t\right) =t,\,0\le t\le 1;$ for $n>2$ the points $q_{1},\ldots ,q_{n-1}$ divide the interval $\left[ 0,1\right]$ into $n-2$ subintervals, and if $\left[ q_{i},q_{j}\right]$ is the subinterval which contains the point $q_{n}$, then $e_{n}\left( t\right) =0$ for $t\in [0, q_{i} ] \cup [ q_{j}, 1 ],\,e_{n}\left( q_{n}\right) =1,$ and $e_{n}$ is a linear function in the interval $\left[ q_{i},q_{n} \right]$ as well as in $\left[ q_{n},q_{j}\right]$. It is known that the described system of Schauder hat functions forms a basis for the space $C\left[ 0,1\right]$ of real-valued continuous functions in $\left[ 0,1\right]$ with the supremum norm

$$\begin{aligned} \left\| f\right\| _{\infty } =\sup _{0\le t\le 1}\left| f\left( t\right) \right| , \end{aligned}$$

see, e.g., [10, Prop. 2.3.5, p. 29].

Now with every sequence $x=\left( x_{1},x_{2},\ldots \,\right) \subset \mathbb {R}^{ \mathbb {N}}$, we associate an element g having coefficients $\left( x_{1},x_{2},\ldots \,\right)$ in the basis $\left\{ e_{n}\left( t\right) ,n\ge 1\right\} ,$ formally written as $g:=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right)$, and define

$$\begin{aligned} \left\| g\right\|= & {} \left\| \sup \left\{ \left| x_{1}e_{1}+x_{2}e_{2}\right| ,\left| x_{1}e_{1}-x_{2}e_{2}\right| , \ldots ,\left| x_{1}e_{1}+\cdots +x_{n-1}e_{n-1}+x_{n}e_{n}\right| ,\right. \right. \\&\qquad \left. \left. \left| x_{1}e_{1}+\cdots +x_{n-1}e_{n-1}-x_{n}e_{n}\right| ,\ldots \right\} \right\| _{\infty } \\= & {} \left\| \sup \limits _{n\ge 1} \left\{ \left| x_{1}e_{1}+\cdots +x_{n}e_{n}\right| +\left| x_{n+1}e_{n+1}\right| \right\} \right\| _{\infty }, \end{aligned}$$

along with

$$\begin{aligned} S_{b}\left[ 0,1\right] =\left\{ g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right) {:}\,\left\| g\right\| <\infty \right\} . \end{aligned}$$

Then $\left( S_{b}\left[ 0,1\right] ,\left\| \cdot \right\| \right)$ is a (nonseparable) Banach space. The space $S_{b}\left[ 0,1\right]$ consists of elements $g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right)$ with finite sums of functions $x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{n}e_{n}$ bounded uniformly in $0\le t\le 1$ and $n\ge 1$, but not necessarily convergent function series $x_{1}e_{1}+x_{2}e_{2}+\cdots$

Furthermore, denote

$$\begin{aligned} S_{c}\left[ 0,1\right] =&\left\{ g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \in S_{b}\left[ 0,1\right] {:}\,\text {the series} \sum \limits _{n}x_{n}e_{n} \right. \\&\left. \qquad \text {converges in norm }\left\| \cdot \right\| \right\} . \end{aligned}$$

It can be shown that $S_{c} [0,1]$ as a set of functions is identically equal to C[0, 1], and on account of the well-known open mapping theorem, the norm $\left\| \cdot \right\|$ is equivalent to $\Vert \cdot \Vert _{\infty }$; therefore, $\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right)$ treated as a function Banach space with its norm topology is the same as $\left( C[0,1], \Vert \cdot \Vert _{\infty }\right)$. Since finite linear combinations $\sum \nolimits _{k=1}^{n}w_{k}e_{k}$ with rational coefficients form a countable dense set in C[0, 1], the space $S_{c}\left[ 0,1\right]$ is separable. The Schauder basis $\left\{ e_{n} ( t ), n\ge 1\right\}$ in $\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right)$ is monotone, because finite linear combinations of hat functions $e_{n} ( t ),\,n\ge 1,$ are piecewise linear with an increasing number of nodes. However, the basis $\left\{ e_{n} ( t ), n\ge 1\right\}$ is not boundedly complete, for bounded finite linear combinations of basic functions need not define a conditionally uniformly convergent series of functions. Moreover, the basis $\left\{ e_{n} ( t ), n\ge 1\right\}$ in $S_{c}\left[ 0,1\right]$ is quasi-orthogonal with respect to $\left\| \cdot \right\|$, but it is not quasi-orthogonal in $\left( C[0,1], \Vert \cdot \Vert _{\infty }\right)$, and it is not unconditional. More precisely, properties of a given Schauder basis in C[0, 1] depend on the shape of the unit sphere, and from our considerations it follows that for each Schauder basis $\left\{ e_{n} ( t ), n\ge 1\right\}$ there can be defined a norm $\left\| \cdot \right\|$ equivalent to the original supremum norm $\Vert \cdot \Vert _{\infty }$ in C[0, 1], such that the basis $\left\{ e_{n}\left( t\right) ,n\ge 1\right\}$ becomes quasi-orthogonal with respect to $\left\| \cdot \right\|$, although the same basis need not be quasi-orthogonal with respect to $\Vert \cdot \Vert _{\infty }$.

By analogy to $S_{c,\epsilon }$ one can now define the spaces $S_{c,\epsilon }\left[ 0,1\right]$ consisting of function series $\sum _n\epsilon _n x_n e_n$ convergent (conditionally) uniformly in $0\le t \le 1$ with norms

$$\begin{aligned} \left\| \left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \right\| _{\epsilon } =\left\| \left( \epsilon _{1}x_{1}e_{1}+\epsilon _{2}x_{2}e_{2}+\cdots \right) \right\| , \end{aligned}$$

for all sequences of signs $\epsilon =\left( \epsilon _{1},\epsilon _{2},\ldots \right) \in \left\{ -1,1\right\} ^{\mathbb {N}}.$ The intersection

$$\begin{aligned} \bigcap \limits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }\left[ 0,1\right] :=\ell ^{1}\left[ 0,1\right] \end{aligned}$$

is then the Banach space of function series convergent absolutely uniformly in $0\le t\le 1,$ with the norm

$$\begin{aligned} \sup \left\{ \left\| \left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \right\| _{\epsilon }{:}\,\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}\right\} =\sup \limits _{0\le t\le 1}\sum \limits _{n=1}^{\infty }\left| x_{n}e_{n}\left( t\right) \right| . \end{aligned}$$

The last formula follows from the fact that Schauder hat functions $e_n(t),\,n\ge 1$, are nonnegative, and in a more general case this is a consequence of a theorem by Sierpiński, cf. [10, Prop. 1.5.7, p. 19]. Clearly, the basis $\{ e_n(t), n\ge 1 \}$ in $\ell ^{1}\left[ 0,1\right]$ is quasi-orthogonal, monotone, unconditional and boundedly complete; thus, $\ell ^{1}\left[ 0,1\right]$ possesses the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64 and Ch. III, §3, Corollary 8, p. 83].

To estimate the rate of (uniform) convergence of function series of the form $g\left( t\right) =\sum \nolimits _{n=1}^{\infty }x_{n}e_{n}\left( t\right) ,\,t\in \left[ 0,1\right] ,$ we may introduce various norms similar as in $\ell ^{p},\, 1\le p<\infty ;$ namely, let $\left( R_{n}g\right) \left( t\right) =\sum \nolimits _{k\ge n}x_{k}e_{k} ( t ),$ and let

$$\begin{aligned} S_{c}^{p}\left[ 0,1\right] =\left\{ g =\sum \limits _{n=1}^{\infty }x_{n}e_{n}\left( t\right) \in S_{c}[0,1]{:}\, \left\| g\right\| _{p} =\left( \sum \limits _{~n=1}^{\infty }\left\| \left( R_{n}g\right) \right\| ^{p}\right) ^{1/p} <\infty \right\} . \end{aligned}$$

Arguing as in example 2, we conclude that the basis $\left\{ e_{n} ( t), n\ge 1\right\}$ in the space $\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right)$ is quasi-orthogonal, but it is not unconditional. Moreover, the basis $\left\{ e_{n} (t), n\ge 1\right\}$ in $\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right)$ is monotone and boundedly complete; thus, by Dunford’s theorem the spaces $\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right)$ possess the RNP, see [4, Ch. III, §1, Th. 6, p. 64].

Remark 1

The same idea as above leads to other examples of Banach spaces of a similar kind. For example, let $I=\left\{ i_{n},n\ge 1\right\}$ be a sequence of positive integers such that $1\le i_{n}\le n,\,i_{n}\nearrow \infty$ and $n-i_{n}\nearrow \infty$ as $n\rightarrow \infty ,$ and for $x=\left\{ x_{n}\right\} \in \mathbb {R}^{\infty },$ let

$$\begin{aligned} \left\| x\right\| _{I,p}=\left\{ \sum \limits _{n=1}^{\infty }\left( \left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right| \right) ^{p}\right\} ^{1/p}, \end{aligned}$$

where $1\le p<\infty ,$ and

$$\begin{aligned} \left\| x\right\| _{I,\infty }=\sup \limits _{n\ge 1}\left\{ \left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right| \right\} . \end{aligned}$$

Define

$$\begin{aligned} \mathbb {B}_{I,p}=\left\{ x=\left\{ x_{n}\right\} \in \mathbb {R}^{\infty }{:}\, \text { the series }\sum \limits _{n}x_{n}e_{n}\text { converges in norm } \left\| \cdot \right\| _{I,p}\,\right\} , \end{aligned}$$

$1\le p\le \infty .$ Then $\mathbb {B}_{I,p},\,1\le p\le \infty ,$ are Banach spaces such that $\left\{ e_{n},n\ge 1\right\}$ is a quasi-orthogonal basis with respect to $\left\| \cdot \right\| _{I,p},$ but in general not unconditional. Moreover, for $1\le p<\infty$ the basis $\left\{ e_{n},n\ge 1\right\}$ in $\mathbb {B}_{I,p}$ is boundedly complete, and thus, these spaces possess the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64]. Next, if $I^{\left( 1\right) }=\left\{ i_{n}^{\left( 1\right) },n\ge 1\right\} ,\,I^{\left( 2\right) }=\left\{ i_{n}^{\left( 2\right) },n\ge 1\right\} , \ldots , I^{\left( k\right) }=\left\{ i_{n}^{\left( k\right) },n\ge 1\right\}$ are some sequences of positive integers satisfying conditions: $1\le i_{n}^{\left( 1\right) }<\cdots<i_{n}^{\left( k\right) }\le n,\,i_{n}^{\left( j\right) }\nearrow \infty ,\,1\le j\le k,\, i_{n}^{\left( j+1\right) }-i_{n}^{\left( j\right) }\nearrow \infty ,\,1\le j<k,$ and $n-i_{n}^{\left( k\right) }\nearrow \infty$ as $n\rightarrow \infty ,$ then for $n-i_{n}\ge k\ge 2$ (replacing $i_{n}$ by $i_{n}^{\left( 1\right) }$), the sums $\left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right|$ can be divided into blocks of the form

$$\begin{aligned} \left| x_{i_{n}^{\left( 1\right) }}+\cdots +x_{i_{n}^{\left( 2\right) }-1}\right| +\left| x_{i_{n}^{\left( 2\right) }}+\cdots +x_{i_{n}^{\left( 3\right) }-1}\right| +\cdots +\left| x_{i_{n}^{\left( k\right) }}+\cdots +x_{n}\right| +\left| x_{n+1}\right| ; \end{aligned}$$

in addition, for $i\ge 1$, instead of $x_i e_i$, the terms $\epsilon _i x_i e_i$ can be used, where $\{ \epsilon _i, i\ge 1 \}$ are various sequences of signs $\pm \,1$. By means of these expressions, in an analogous way as above, new norms and new Banach spaces with quasi-orthogonal bases can be defined.

4 Convergence of the Banach Random Walk

According to the standard terminology, the measure G on the Borel $\sigma$-field $\mathcal { B}\left( \mathbb {R}\right)$ is called here symmetric, if $G\left( -A\right) =G\left( A\right)$ for all $A\in \mathcal {B}\left( \mathbb {R}\right)$.

Lemma 1

Let $\left\{ b_{n},n\ge 1\right\}$ be a quasi-orthogonal Schauder basis in a real Banach space $\left( \mathbb {B} ,\left\| \cdot \right\| \right)$, and let $\left\{ G_{n},n\ge 1\right\}$ be a sequence of symmetric probability distributions concentrated on the interval $\left[ -\,1,1\right] \subset \mathbb {R}$. Then the BRW $\left\{ Z_{n},n\ge 1\right\}$ is a $\mathbb {B}$-valued martingale with respect to the natural filtration $\mathcal {F}_{n}=\sigma \left( X_{1},X_{2},\ldots ,X_{n}\right) =\sigma \left( Z_{1},Z_{2},\ldots ,Z_{n}\right) ,\, n\ge 1$.

Proof

From the construction of BRW in a Banach space, it follows that $Z_{n}\in B$ for all $n\ge 1$, i.e., the r.e.’s $Z_{n}$ are bounded and therefore Bochner integrable. Moreover, each $Z_{n}$ is $\mathcal {F}_{n}$-measurable; thus,

$$\begin{aligned} E\left[ Z_{n+1}|\mathcal {F}_{n}\right] =E\left[ Z_{n}+X_{n+1}b_{n+1}| \mathcal {F}_{n}\right] =Z_{n}+E\left[ X_{n+1}|\mathcal {F}_{n}\right] b_{n+1} ~\, \text {a.s.} \end{aligned}$$

Hence it suffices to show the equality $E\left[ X_{n+1}|\mathcal {F}_{n}\right] =0$ a.s. But the last statement is obvious, since it is known that $X_{n+1}$ possesses a symmetric distribution in the a.s. symmetric interval $\left[ \alpha _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right] =\left[ -\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right]$. $\square$

Lemma 2

Let $\left\{ Z_{n},n\ge 1\right\}$ be the BRW in a Banach space $\mathbb {B}$ with a quasi-orthogonal Schauder basis $\left\{ b_{n},n\ge 1\right\}$ and let $\left\{ \mathcal {F}_{n},n\ge 1\right\}$ be the defined above filtration associated with $\left\{ Z_{n},n\ge 1\right\} .$ Denote by $\mathcal {F}_{\infty }$ the $\sigma$-field generated by the field $\bigcup _{n=1}^{\infty }\mathcal {F}_{n}$, i.e., $\mathcal {F}_{\infty }=\sigma \left( \bigcup _{n=1}^{\infty } \mathcal {F}_{n}\right)$. Then the following statements are true:

• (a) There exists a vector measure $\upsilon {:}\,\mathcal {F}_{\infty }\rightarrow \mathbb {B}$ with bounded variation, absolutely continuous with respect to P,  such that

  $$\begin{aligned} \left\langle Z_{n},x^{*}\right\rangle \rightarrow \frac{\mathrm{d}\left\langle \upsilon ,x^{*}\right\rangle }{\mathrm{d}P} ~ \text { a.s. for all } x^{*}\in \mathbb {B}^{*}, \end{aligned}$$

• (b) If there is a r.e. $\xi \in L^{1}\left( \mathcal {F}_{\infty };\mathbb {B} \right)$ such that

$$\begin{aligned} \left\langle Z_{n},x^{*}\right\rangle \rightarrow \left\langle \xi ,x^{*}\right\rangle ~ \text { a.s.} \end{aligned}$$

for each $x^{*}\in \mathbb {B}^{*}$, then

$$\begin{aligned} \Vert Z_{n}-\xi \Vert \rightarrow 0 ~ \text { a.s.} \end{aligned}$$

Proof

Since the BRW $\left\{ Z_{n},n\ge 1\right\}$ satisfies the condition $Z_{n}\in B,\,n\ge 1$, we have $\sup _{n\ge 1}E\left\| Z_{n}\right\| \le 1<\infty .$ Therefore, our result is a direct consequence of a theorem given by Stegall, which can be found in [11, Ch. II, §4.3, Prop. 4.3, p. 132]. $\square$

Lemma 3

For each set $A\in \mathcal {F}_{\infty }$ there exists

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A}Z_{n}\mathrm{d}P=V\left( A\right) \end{aligned}$$

in the strong topology of $\mathbb {B}$, and the mapping $V{:}\,\mathcal {F} _{\infty }\rightarrow \mathbb {B}$ is a countably additive vector measure.

Proof

Observe first that in view of Jensen’s inequality for conditional expectations in a Banach space, the sequence $\left\{ \left\| Z_{n}\right\| ,\mathcal {F} _{n},n\ge 1\right\}$ is a real-valued submartingale, cf. [11, Ch. II, §4.1, (g), p. 127], or [12]. Furthermore,

$$\begin{aligned} \sup _{n\ge 1}E\left\| Z_{n}\right\| ^{p}\le 1<\infty ~ \text { for each } 1\le p<\infty , \end{aligned}$$

so that r.v.’s $\left\{ \left\| Z_{n}\right\| ,n\ge 1\right\}$ are uniformly integrable, which implies a.s. convergence $\left\| Z_{n}\right\| \rightarrow Z_{\infty }$ (and in $L^{1}$), where $Z_{\infty }\in L^{p}=L^{p}(\mathbb {R})$ for every fixed $1\le p<\infty$, see, e.g., [8, Ch. IV, Th. IV-1-2, p. 62, and Prop. IV-5-24, p. 91]. In particular,

$$\begin{aligned} \int \nolimits _{A}\left\| Z_{n}\right\| \mathrm{d}P\rightarrow \int \nolimits _{A} Z_{\infty } \mathrm{d}P \end{aligned}$$

for each measurable set $A\in \mathcal {F}$. Next, if $B\in \bigcup \nolimits _{n}\mathcal {F} _{n}$, then by the martingale property of $\left\{ Z_{n},n\ge 1\right\} ,$

$$\begin{aligned} \int \nolimits _{B}Z_{n}\mathrm{d}P\rightarrow V\left( B\right) \end{aligned}$$

strongly in $\mathbb {B}$. Let $\varepsilon >0$ be arbitrary and let $\delta >0$ be chosen in such a way that $\int \nolimits _{C}Z_{\infty } \mathrm{d}P<\varepsilon /3$ whenever $C\in \mathcal {F}_{\infty }$ and $P\left[ C\right] <\delta$. Given any set $A\in \mathcal {F}_{\infty },$ select $B\in \bigcup \nolimits _{n} \mathcal {F}_{n}$ satisfying condition $P\left[ A\div B\right] <\delta$. Obviously,

$$\begin{aligned}&\left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| \le \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| \\&\quad +\,\left\| \int \nolimits _{B}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{m}\mathrm{d}P\right\| +\left\| \int \nolimits _{B}Z_{m}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| . \end{aligned}$$

Moreover,

$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| \le \int \nolimits _{A\setminus B}\left\| Z_{n}\right\| \mathrm{d}P+\int \nolimits _{B\setminus A}\left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{A\div B}\left\| Z_{n}\right\| \mathrm{d}P. \end{aligned}$$

Passing to the limit as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A\div B}\left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{A\div B} Z_{\infty } \mathrm{d}P<\varepsilon /3, \end{aligned}$$

thus

$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| <\varepsilon /3 \end{aligned}$$

for sufficiently large $n\ge n_{0}$. Since the sequence $\left\{ \int \nolimits _{B}Z_{n}\mathrm{d}P,n\ge 1\right\}$ is Cauchy in $\mathbb {B}$, we also conclude that

$$\begin{aligned} \left\| \int \nolimits _{B}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{m}\mathrm{d}P\right\| <\varepsilon /3 \end{aligned}$$

for all large enough $m>n\ge n_{1}$. Consequently,

$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| <\varepsilon , \end{aligned}$$

whenever $m>n\ge \max \left\{ n_{0},n_{1}\right\}$. In other words, the sequence of integrals $\left\{ \int \nolimits _{A}Z_{n}\mathrm{d}P,n\ge 1\right\}$ is Cauchy in $\left( \mathbb {B},\left\| \cdot \right\| \right)$, and therefore, there exists

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A}Z_{n}\mathrm{d}P=V\left( A\right) \end{aligned}$$

in the strong topology of $\mathbb {B}$ for each set $A\in \mathcal {F}_{\infty }$. It can be easily seen that $V{:}\, \mathcal {F}_{\infty } \rightarrow \mathbb {B}$ is finitely additive. Let $A_{1},A_{2},\ldots \in \mathcal {F}_{\infty }$ be an arbitrary sequence of pairwise disjoint sets. Notice that

$$\begin{aligned} \left\| \lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=1}^{\infty }A_{k}}Z_{n}\mathrm{d}P-\lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=1}^{m}A_{k}}Z_{n}\mathrm{d}P\right\| =\lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}}Z_{n}\mathrm{d}P\right\| , \end{aligned}$$

and thus, to prove countable additivity of V it is enough to show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}}Z_{n}\mathrm{d}P\right\| \rightarrow 0 \end{aligned}$$

as $m\rightarrow \infty$. Taking $m_{0}$ so large that $P\left[ \bigcup \nolimits _{k=m+1}^{\infty }A_{k}\right] <\delta$ for $m\ge m_{0}$, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} Z_{n}\mathrm{d}P\right\| \le \lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} \left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} Z_{\infty } \mathrm{d}P<\varepsilon /3 \end{aligned}$$

provided $m\ge m_{0}$, which terminates the proof. $\square$

The above Lemma 3 enables us to apply the Lebesgue decomposition theorem for the vector measure V, see [4, Ch. I, §5, Th. 9, p. 31].

Lemma 4

Let

$$\begin{aligned} V\left( A\right) =\lim _{n\rightarrow \infty }\int _{A}Z_{n}\mathrm{d}P, ~\, A\in \mathcal {F}_{\infty }, \end{aligned}$$

and let $V=H+J,\,\left| H\right| \ll P,\,\left| J\right| \perp P,$ be the Lebesgue decomposition of  V with respect to P, where $\left| H\right| ,\,\left| J\right|$ are variations of H and J, respectively. Then $\lim _{n\rightarrow \infty }Z_{n}$ exists a.s. if and only if H has a Radon–Nikodym derivative $h\in L^{1}\left( \mathcal {F};\mathbb {B}\right) .$ Moreover, in this case $\lim _{n\rightarrow \infty }Z_{n}=E\left( h|\mathcal {F}_{\infty }\right)$ a.s.

Proof

Arguing as above, we easily note that $\left\{ Z_{n},\mathcal {F}_{n},n\ge 1\right\}$ is an $L^{1}\left( \mathbb {B}\right)$-bounded martingale (here, and in the sequel $L^{p}\left( \mathbb {B}\right) =L^{p}\left( \varOmega , \mathcal {F}, P ;\mathbb {B}\right) ,\,1\le p < \infty$); thus, the conclusion follows from the martingale pointwise convergence theorem given in [4, Ch. V, §2, Th. 9, p. 130]. $\square$

Lemma 5

Let $\mathbb {B}$ be a Banach space with the RNP and a quasi-orthogonal Schauder basis $\left\{ b_{n},n\ge 1\right\}$. Moreover, let $\left\{ G_{n},n\ge 1\right\}$ be a sequence of symmetric probability distributions concentrated on $\left[ -\,1,1\right] \subset \mathbb {R}$. Then the BRW martingale $\left\{ Z_{n},n\ge 1\right\}$ converges strongly a.s. and in $L^{p}\left( \mathbb {B}\right)$ for each fixed $1\le p<\infty$.

Proof

The limit $\lim _{n\rightarrow \infty }Z_{n}$ of the martingale $\left\{ Z_{n},n\ge 1\right\}$ exists in $L^{p}\left( \mathbb {B}\right)$-norm, if and only if $\sup _{n\ge 1}\left\| Z_{n}\right\| _{p}^{p}=\sup _{n\ge 1}E\left\| Z_{n}\right\| ^{p}<\infty ,$ where $1<p<\infty ,$ which is evident as $Z_{n}\in B$ for $n\ge 1.$ The last observation implies uniform integrability of random elements $\left\{ Z_{n},n\ge 1\right\} ,$ and we have obviously $\sup _{n\ge 1}\left\| Z_{n}\right\| _{1}=\sup _{n\ge 1}E\left\| Z_{n}\right\| \le 1<\infty .$ Thus $\lim _{n\rightarrow \infty }Z_{n}$ exists as well in $L^{1}\left( \mathbb {B}\right)$-norm in view of the martingale mean convergence theorem, cf. [4, Ch. V, §2, Corollary 4, p. 126]. It is also well known that an $L^{1}\left( \mathbb {B}\right)$ convergent martingale converges a.s. to its $L^{1}\left( \mathbb {B}\right)$-limit, see [4, Ch. V, §2, Th. 8, p. 129], or [11, Ch. II, §4.3, Th. 4.2, p. 131 and Th. 4.3, p. 136]. $\square$

Corollary 1

If $\mathbb {B}$ is a Banach space with a quasi-orthogonal boundedly complete Schauder basis $\left\{ b_{n},n\ge 1\right\} ,$ then the BRW $\left\{ Z_{n},n\ge 1\right\}$ in $\mathbb {B}$ converges strongly a.s. and in $L^{p}\left( \mathbb {B}\right)$ for each fixed $1\le p<\infty$.

Proof

By a theorem of Dunford, if a Banach space $\mathbb {B}$ possesses a boundedly complete Schauder basis, then $\mathbb {B}$ has the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64]. Hence and from Lemma 5, the assertion of Corollary 1 follows. $\square$

Corollary 2

Let $\mathbb {B}$ be a reflexive Banach space which has a quasi-orthogonalSchauder basis $\left\{ b_{n},n\ge 1\right\}$. Then the $\mathbb {B}$-valued BRW $\left\{ Z_{n},n\ge 1\right\}$ converges strongly a.s. and in $L^{p}\left( \mathbb {B}\right)$ for each fixed $1\le p<\infty$. In particular, if $\mathbb {B}=\mathbb {H}$ is a Hilbert space with a basis $\left\{ b_{n},n\ge 1\right\}$ which forms a CONS in $\mathbb {H}$, then the last statement remains valid.

Proof

It is fairly well known from a theorem of Phillips that reflexive Banach spaces have the RNP, see [4, Ch. III, §3, Corollary 4, p. 82]. Since each Hilbert space is reflexive, we conclude that $\mathbb {B}=\mathbb {H}$ has the RNP. Thus, an application of Lemma 5 concludes the proof. $\square$

Theorem 1

Let $\phi {:}\,\mathbb {B}\rightarrow \mathbb {R}$ be a bounded and continuous mapping in a Banach space $\mathbb {B}$ which has the RNP and a quasi-orthogonal Schauder basis $\left\{ b_{n},n\ge 1\right\}$. If $\left\{ Z_{n},n\ge 1\right\}$ is the BRW in $\mathbb {B},$ then

$$\begin{aligned} \phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right) ~\, \text {a.s. and in } L^{p}=L^{p}\left( \mathbb {R}\right) ,\quad 1\le p<\infty , \end{aligned}$$

where $\xi =\lim _{n\rightarrow \infty }Z_{n}$ a.s. and in $L^{p}\left( \mathbb {B}\right)$-norm for all $1\le p<\infty .$ In particular, there exists

$$\begin{aligned} \lim _{n\rightarrow \infty }E\phi \left( Z_{n}\right) =E\phi \left( \xi \right) . \end{aligned}$$

Proof

Since $\phi$ is continuous and the assumptions of Lemma 5 are fulfilled, we conclude that $\phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right)$ a.s. But in addition $\phi$ is assumed to be bounded, thus using the Lebesgue-dominated convergence theorem we obtain also convergence $\phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right)$ in $L^{p},\,1\le p<\infty$. The last statement of the theorem follows from the estimate

$$\begin{aligned} \left| E\phi \left( Z_{n}\right) -E\phi \left( \xi \right) \right| \le E\left| \phi \left( Z_{n}\right) -\phi \left( \xi \right) \right| \rightarrow 0. \end{aligned}$$

$\square$

Corollary 3

The assertion of Theorem 1 remains valid for a Banach space $\mathbb {B}$ with a boundedly complete quasi-orthogonal Schauder basis $\left\{ b_{n},n\ge 1\right\} ,$ as well as for a Hilbert space $\mathbb {B}=\mathbb {H}$ with the Schauder basis $\left\{ b_{n},n\ge 1\right\}$ that forms a CONS in $\mathbb {H}$.

5 The Banach Functional Integral in a Class of Banach Spaces

Let $\mathcal {C}_b =\left\{ \phi {:}\,B\rightarrow \mathbb {R};\phi \text {-bounded and continuous}\right\}$. Observe that $\mathcal {C}_b$ has the following properties:

(i$_{1}$) the set $\mathcal {C}_b$ is a real linear space,

(i$_{2}$) if $\phi \in \mathcal {C}_b$, then $\left| \phi \right| \in \mathcal {C}_b$.

Define a functional $f{:}\,\mathcal {C}_b\rightarrow \mathbb {R}$ by the formula:

$$\begin{aligned} f\left( \phi \right) =\lim _{n\rightarrow \infty }E\phi \left( Z_{n}\right) =E\phi \left( \xi \right) , \end{aligned}$$

(4)

where $\left\{ Z_{n},n\ge 1\right\}$ is a BRW in the Banach space $\mathbb { B}$, and $\xi =\lim _{n\rightarrow \infty }Z_{n}$ a.s. and in $L^{p}\left( \mathbb {B}\right) ,\,1\le p<\infty .$

It can be easily seen that the mapping f satisfies the following conditions:

(ii$_{1}$) $f{:}\,\mathcal {C}_b\rightarrow \mathbb {R}$ is a linear functional,

(ii$_{2}$) the functional f is nonnegative, i.e., $f\left( \phi \right) \ge 0$ whenever $\phi \in \mathcal {C}_b$ and $\phi \ge 0$,

(ii$_{3}$) if 1$^{0}\,\left\{ \phi _{n}\right\} \subset \mathcal {C }_b,\,\psi \in \mathcal {C}_b$, 2$^{0}\,\left| \phi _{n}\right| \le \psi$ for $n\ge 1$, and 3$^{0}\,\lim _{n\rightarrow \infty }\phi _{n}\left( x\right) =0$ for all $x\in B$, then $\lim _{n\rightarrow \infty }f\left( \phi _{n}\right) =0$.

Notice that 3$^{0}$ implies P-a.s. convergence $\phi _{n}\left( \xi \right) \rightarrow 0$; thus, the last condition follows from the classical Lebesgue-dominated convergence theorem applied to integrals $E\phi _{n}\left( \xi \right) ,\,n=1,2,\ldots$ (Actually, in our approach we can even replace condition 3$^{0}$ by a weaker assumption $\phi _{n} \rightarrow 0$ in $P\circ \xi ^{-1}$-measure.) Consequently, the functional f satisfies all the conditions given in §2 of the Banach paper [1]. Therefore, for our functional f the Banach Th. 1, §3, p. 322, [1] is valid. In this way, we obtain the following result.

Theorem 2

Let $\left\{ Z_{n},n\ge 1\right\}$ be a BRW in a Banach space $\mathbb {B}$ with the RNP and a quasi-orthogonal Schauder basis $\left\{ b_{n},n\ge 1\right\} ,$ in particular in a Banach space $\mathbb {B}$ with a boundedly complete quasi-orthogonal Schauder basis. Then, the functional f given by (4) has an extension to the additive functional F on the linear set $\mathcal {L}\supset \mathcal {C}_b$ of all bounded, Borel measurable functions $\varPhi {:}\, B\rightarrow \mathbb {R}$. Moreover, the extended functional F on $\mathcal {L}$ possesses all the properties (A)–(E) and (R) specified in §1 of the Banach paper [1], analogous to the Lebesgue integral.

Remark 2

The approach presented above is a generalization of the method proposed by Banach [1] for the construction of the so-called $\mathfrak {L}$-integral—an analogue of the Lebesgue integral in abstract spaces.

Theorem 3

Let $\mathbb {B}$ be a Banach space with the RNP and a quasi-orthogonal Schauder basis, in particular—a Banach space with a boundedly complete quasi-orthogonal Schauder basis. Then each sequence of symmetric probability distributions $\left\{ G_{n},n\ge 1\right\}$ concentrated on the interval $[-\,1,1]\subset \mathbb {R}$ generates a probability measure $\varGamma$ on the Borel $\sigma$-field $\mathcal {B}$ in $\mathbb {B},$ given by

$$\begin{aligned} \varGamma \left( A\right) =E\mathbb {1}_{A}\left( \xi \right) , ~ A\in \mathcal {B}. \end{aligned}$$

The measure $\varGamma$ is equal to the limit distribution of the described above BRW $\left\{ Z_{n},n\ge 1\right\}$ in $\mathbb {B}$, thus ${\mathrm{supp}\,}\varGamma \subseteq B.$

Proof

Obviously, $\varGamma$ is nonnegative and normalized so that $\varGamma \left( \mathbb {B}\right) =\varGamma \left( B\right) =1$. It suffices to verify countable additivity of $\varGamma$, but it follows immediately from the properties of the integral $E\left( \cdot \right)$. The last conclusion can also be easily shown in a direct way. To this end, let $A_{1},A_{2},\ldots \in \mathcal {B}$ be arbitrary disjoint sets. Since $\mathrm {supp\,}\varGamma \subseteq B,$ we have $\varGamma \left( A \right) =\varGamma \left( A \cap B\right) =E\mathbb {1}_{A \cap B}\left( \xi \right) ,\,A \in \mathcal {B}$. Observe next that

$$\begin{aligned} E\mathbb {1} _{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) = E\left( \sum \limits _{j=1}^{n}\mathbb {1}_{A_{j}\cap B}\left( \xi \right) \right) =\sum \limits _{j=1}^{n}E\mathbb {1}_{A_{j}\cap B}\left( \xi \right) =\sum \limits _{j=1}^{n}\varGamma \left( A_{j}\right) , \end{aligned}$$

and

$$\begin{aligned} 0\le \sum \limits _{j=1}^{n}\mathbb {1}_{A_{j}\cap B}\left( x\right) = \mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( x\right) \nearrow \mathbb {1}_{\bigcup \nolimits _{n=1}^{\infty }\left( A_{n}\cap B\right) }\left( x\right) \le \mathbb {1}_{B}\left( x\right) ,~ x\in B. \end{aligned}$$

Hence, on account of the Lebesgue monotone convergence theorem,

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\varGamma \left( A_{n}\right)= & {} \lim _{n\rightarrow \infty }\sum \limits _{j=1}^{n}\varGamma \left( A_{j}\right) =\lim _{n\rightarrow \infty }E\mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) \\= & {} E\left( \lim _{n\rightarrow \infty }\mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) \right) =E\mathbb {1}_{\left( \bigcup \nolimits _{n=1}^{\infty } A_{n}\right) \cap B }\left( \xi \right) =\varGamma \left( \bigcup \limits _{n=1}^{\infty }A_{n}\right) . \end{aligned}$$

$\square$

From the construction of the BRW in a Banach space, it follows immediately that the limit distribution $\varGamma =P\circ \xi ^{-1}$ of the BRW is sign-invariant with respect to the Schauder basis $\{ b_{n}, n\ge 1\}$, in the sense that for each set $A\in \mathcal {B}$ and every sequence $\epsilon =\{\epsilon _{1},\epsilon _{2},\ldots \}$ of signs $\epsilon _{k} \in \{-1,1\},\,k\ge 1$, we have

$$\begin{aligned} \varGamma (A) = \varGamma (\epsilon A), \end{aligned}$$

where $\epsilon A= \left\{ \sum _{k=1}^{\infty } \epsilon _{k} x_{k} b_{k} \in \mathbb {B} {:}\, \sum _{k=1}^{\infty } x_{k} b_{k} \in A\right\}$. It is also clear that each sign-invariant measure is symmetric, thus

$$\begin{aligned} \varGamma (A) = \varGamma (-A) ~\, \text {for all} ~ A\in \mathcal {B}. \end{aligned}$$

By analogy to the notion of the Wiener measure, we propose to call $\varGamma$the Banach measure in a Banach space. One may expect that the Banach measure will play a similarly important role in Banach spaces as is played by the Gaussian measure constructed by Gross [5], cf. Bogachev [3], or Kuo [6].

6 Examples

• 1. Let $\mathbb {B}=\ell ^{p},\,1\le p < \infty$, and let $b_n = e_n = (0,\ldots ,0,1,0,\ldots ),\,n\ge 1$, where 1 is the nth term of the sequence $(0,\ldots ,0,1,0,\ldots )$. Then $\pi _n(x) = \sum _{k=1}^{n} x_k b_k = (x_{1},\ldots ,x_{n},0,\ldots )$ for $x=\sum _{n=1}^{\infty }x_{n}b_{n} = (x_{1},x_{2},\ldots ) \in \ell ^{p}$, and thus

  $$\begin{aligned} \left| \pi _{n}(x)+tb_{n+1}\right| _{p}^{p}=\sum \limits _{k=1}^{n}|x_{k}|^{p}+|t|^{p}\le 1 ~ \Leftrightarrow ~ |t|\le \left( 1-\sum \limits _{k=1}^{n}|x_{k}|^{p}\right) ^{1/p}. \end{aligned}$$

Hence it follows that $\{b_{n},n\ge 1\}$ is a quasi-orthogonal Schauder basis in $\ell ^{p}$. (In fact, the considered basis is unconditional.) Moreover, if

$$\begin{aligned} \sup _{n\ge 1}\left| \pi _{n}(x)\right| _{p}=\sup _{n\ge 1}\left| \sum \limits _{k=1}^{n}x_{k}b_{k}\right| _{p}=\left( \sum \limits _{k=1}^{\infty }|x_{k}|^{p}\right) ^{1/p}\le M<\infty , \end{aligned}$$

then the series $\sum _{n=1}^{\infty }x_{n}b_{n}$ converges in $\ell ^{p}$. Thus, the basis $\{b_{n},n\ge 1\}$ is boundedly complete and in consequence each space $\ell ^{p},\,1\le p < \infty$, has the RNP. It is also well known that for $p >1$ the spaces $\ell ^p$ are reflexive, which implies as well that they have the RNP. Therefore, all the above results are valid for Banach spaces $\mathbb {B}=\ell ^{p},\,1\le p < \infty$.

• 2. Let $\mathbb {B}=L^{p}\left[ 0,1\right] ,$ where $1\le p<\infty .$ Consider the system of Haar functions: $h_{1}^{0}\left( s\right) =1 ,\,s\in \left[ 0,1\right]$, and

  $$\begin{aligned} h_{k}^{n}\left( s\right) = 2^{n}\cdot \mathbb {1}_{[\left( 2k-2\right) /2^{n+1},\,\left( 2k-1\right) /2^{n+1})}\left( s\right) - 2^{n}\cdot \mathbb {1}_{[\left( 2k-1\right) /2^{n+1},\,\left( 2k\right) /2^{n+1})}\left( s\right) \end{aligned}$$

for $k=1,2,\ldots ,2^{n},\,n=1,2,\ldots ,\,s\in \left[ 0,1\right]$. It is known that the system of Haar functions forms a Schauder basis in $L^{p}\left[ 0,1\right]$, see, e.g., [9, Th. 24.17, pp. 290–295], or [7, Part II, Prop. 2.c.1, p. 150]. For convenience of the reader, we sketch here the proof that the Haar basis in $L^{p}\left[ 0,1\right]$ is quasi-orthogonal.

Proposition 1

The system of Haar functions is a quasi-orthogonal basis in $L^{p}\left[ 0,1\right] ,\,1\le p<\infty$.

Proof

Let the Haar functions be arranged in a sequence that is divided into blocks, each of $2^{n}$ members, numbered by upper indices $n=0,1,2,\ldots ,$

$$\begin{aligned} \varLambda =\left\{ \left( h_{1}^{0}\right) ,\left( h_{1}^{1},h_{2}^{1}\right) ,\left( h_{1}^{2},h_{2}^{2},h_{3}^{2},h_{4}^{2}\right) ,\left( h_{1}^{3},h_{2}^{3},h_{3}^{3},h_{4}^{3},h_{5}^{3},h_{6}^{3},h_{7}^{3},h_{8}^{3}\right) ,\ldots \right\} . \end{aligned}$$

We make now two crucial observations: 1$^{0}$ for a fixed $n\ge 0$ within the same nth block the Haar functions $h_{k}^{n},\, 1\le k\le 2^{n}$, have nonoverlapping supports, 2$^{0}$ the function of the form $c_{1}^{0}h_{1}^{0}+c_{1}^{1}h_{1}^{1}+c_{2}^{1}h_{2}^{1}+\cdots +c_{1}^{n-1}h_{1}^{n-1}+ \cdots +c_{2^{n-1}}^{n-1}h_{2^{n-1}}^{n-1},$ where $c_{j}^{i}\in \mathbb {R}$ are arbitrarily fixed coefficients, is constant in each interval of the form $\left[ \left( k-1\right) /2^{n},k/2^{n}\right) ,\,k=1,2,\ldots ,2^{n}.$

Suppose now that $\pi _{k-1}^{n}\left( x\right) =x_{1}^{0}h_{1}^{0}+x_{1}^{1}h_{1}^{1}+x_{2}^{1}h_{2}^{1}+\cdots +x_{k-1}^{n}h_{k-1}^{n}$ and the next element of $\varLambda$ is $h_{k}^{n}.$ Then for any parameter $t\in \mathbb {R},$

$$\begin{aligned} \left\| \pi _{k-1}^{n}\left( x\right) +th_{k}^{n}\right\| _{p}^{p}= & {} \int \nolimits _{0}^{1}\left| \pi _{k-1}^{n}\left( x\right) \left( s\right) +th_{k}^{n}\left( s\right) \right| ^{p}\mathrm{d}s=\int \nolimits _{0}^{\left( k-1\right) /2^{n}}\left| \pi _{k-1}^{n}\left( x\right) \left( s\right) \right| ^{p}\mathrm{d}s\\&+\,\int \nolimits _{\left( k-1\right) /2^{n}}^{k/2^{n}}\left| \pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) +th_{k}^{n}\left( s\right) \right| ^{p}\mathrm{d}s\\&+\,\int \nolimits _{k/2^{n}}^{1}\left| \pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) \right| ^{p}\mathrm{d}s=I_{1}+I_{2}+I_{3}, \end{aligned}$$

where $\pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) = \pi _{2^{n-1}}^{n-1}\left( x\right) \left( (2k-1)/2^{n+1}\right) =c ,\,s\in \big [ (k-1)/{2^{n}}, k/{2^{n}}\big ).$ The first and third integrals on the right-hand side do not depend on the parameter t,  and the middle term is equal to

$$\begin{aligned} I_{2}= & {} \int \nolimits _{\left( k-1\right) /2^{n}}^{\left( 2k-1\right) /2^{n+1}}\left| c+t2^{n}\right| ^{p}\mathrm{d}s+\int \nolimits _{\left( 2k-1\right) /2^{n+1}}^{k/2^{n}}\left| c-t2^{n}\right| ^{p}\mathrm{d}s\\= & {} \left| c+t2^{n}\right| ^{p}\cdot \frac{1}{2^{n+1}}+\left| c-t2^{n}\right| ^{p}\cdot \frac{1}{2^{n+1}}:=r\left( t\right) . \end{aligned}$$

Since $r\left( t\right) =r\left( -t\right)$, and

$$\begin{aligned} \inf \left\{ t\in \mathbb {R}{:}\,r\left( t\right) \le 1-I_{1}-I_{3}\right\}= & {} -\sup \left\{ -t\in \mathbb {R}{:}\,r\left( -t\right) \le 1-I_{1}-I_{3}\right\} \\= & {} -\sup \left\{ t^{\prime }\in \mathbb {R}{:}\,r\left( t^{\prime }\right) \le 1-I_{1}-I_{3}\right\} , \end{aligned}$$

we conclude that $\alpha _{k}^{n}=-\beta _{k}^{n}$. The same argument remains valid when $\pi _{k-1}^{n}$ is replaced by $\pi _{2^{n-1}}^{n-1},$ and $\pi _{k}^{n}$ is replaced by $\pi _{1}^{n},$ and thus, the system of Haar functions forms a quasi-orthogonal basis in $L^{p}\left[ 0,1\right] .$$\square$

Remark 3

It is clear that the Haar functions for $n\ge 1$ can be modified as follows:

$$\begin{aligned} h_k^n\left( s\right) = 2^{n/p}\cdot \mathbb {1}_{[\left( 2k-2\right) /2^{n+1},\,\left( 2k-1\right) /2^{n+1})}\left( s\right) - 2^{n/p}\cdot \mathbb {1}_{[\left( 2k-1\right) /2^{n+1},\,\left( 2k\right) /2^{n+1})}\left( s\right) , \end{aligned}$$

$k=1,2,\ldots ,2^{n},\,s\in \left[ 0,1\right]$. Then the above Proposition 1 for modified Haar functions remains true, and in addition, we have $\Vert h_{k}^{n} \Vert _{p}= 1$ for all k, n.

Evidently, all the spaces $L^{p}\left[ 0,1\right] ,\,1<p<\infty$, are reflexive Banach spaces, and thus, they possess the RNP, which is a straightforward consequence of Phillips’ theorem, cf. [4, Ch. III, §3, Corollary 6, p. 82]. Unfortunately, the space $L^{1}\left[ 0,1\right]$ does not have the RNP, see [4, Ch. VII, p. 219].

Hence, it follows that all the results presented in previous sections are valid for Banach spaces $L^{p}[0,1],\,1<p<\infty$.

Remark 4

Proposition 1 together with observation that the space $L^{1}[0,1]$ does not possess the RNP implies the following conclusion: the existence of a quasi-orthogonal Schauder basis in a Banach space is not a sufficient condition for the RNP. The same conclusion follows from the fact that the space $\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right)$ (isometrically isomorphic to $\left( C[0,1], \Vert \cdot \Vert _{\infty }\right)$) does not have the RNP.

3. To illustrate the technique of computations of Banach $\mathfrak {L}$-integrals based on the method described here, we calculate, for instance, two “rarefied” absolute pth moments of $\xi =\sum _{n=1}^{\infty }X_{n}b_{n}$ in $\ell ^{p},\,1\le p<\infty$, determined by lacunary series $\sum _{n=1}^{\infty } X_{2n-1} b_{2n-1}$ and $\sum _{n=1}^{\infty } X_{2n} b_{2n}$, namely $E\left( \sum _{n=1}^{\infty }\left| X_{2n-1}\right| ^{p}\right)$ and $E\left( \sum _{n=1}^{\infty }\left| X_{2n}\right| ^{p}\right)$, where $\xi$ is the limit random element of the BRW generated by the sequence of identical uniform distributions on $\left[ -\,1,1\right] \subset \mathbb {R}$. Since the basis $\{ b_n, n\ge 1 \}$ in $\ell ^{p}$ is unconditional, the lacunary series considered here are strongly convergent, see, e.g., [7, Prop. 1.c.1, p. 15, and p. 19].

Observe first that if $\pi _\mathrm{odd}\left( x\right) =\sum _{n=1}^{\infty }x_{2n-1}b_{2n-1}$ for $x=\sum _{n=1}^{\infty }x_{n}b_{n}\in \ell ^{p},$ then $\left| \pi _\mathrm{odd}\left( x\right) \right| _{p}=\left( \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p},$ thus

$$\begin{aligned}&\left| \left( \sum \limits _{~n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p}-\left( \sum \limits _{n=1}^{\infty }\left| x_{2n-1}^{\prime }\right| ^{p}\right) ^{1/p}\right| =\left| \left| \pi _\mathrm{odd}\left( x\right) \right| _{p}-\left| \pi _\mathrm{odd}\left( x^{\prime }\right) \right| _{p}\right| \\&\quad \le \left| \pi _\mathrm{odd}\left( x-x^{\prime }\right) \right| _{p}=\left( \sum \limits _{~n=1}^{\infty }\left| x_{2n-1}-x_{2n-1}^{\prime }\right| ^{p}\right) ^{1/p}\le \left| x-x^{\prime }\right| _{p}\rightarrow 0 \end{aligned}$$

as $x\rightarrow x^{\prime }$ in $\ell ^{p}$-norm $\left| \cdot \right| _{p}$. It follows that $\ell ^{p}\ni x\mapsto \left( \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p}$ is a continuous function. Therefore, $\ell ^{p}\ni x\mapsto \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p},$ as well as $\ell ^{p}\ni x\mapsto \sum _{n=1}^{\infty }\left| x_{2n}\right| ^{p}$ are continuous, and both these maps are bounded in the unit ball $B \subset \ell ^{p}$.

According to the construction of the BRW in $\ell ^{p},$ the density of $X_1$ is equal to $f_1(x_1)=\mathbb {1}_{[-1,1]} (x_1)/2$, and the density of $\left( X_{1},X_{2},\ldots ,X_{n}\right)$ for $n>1$ is given by

$$\begin{aligned}&f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \\&\quad =\frac{\mathbb {1}_{K_{n}}\left( x_{1},x_{2},\ldots ,x_{n}\right) }{2^{n}\cdot \left( \left[ 1-\left| x_{1}\right| ^{p}\right] \cdot \left[ 1-\left( \left| x_{1}\right| ^{p}+\left| x_{2}\right| ^{p}\right) \right] \cdot \ldots \cdot \left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] \right) ^{1/p}}, \end{aligned}$$

where $K_{n}=K_{n}\left( 0,1\right)$ is the unit ball with center zero and radius 1 in $\mathbb {R}^{n}$, equipped with the $\ell ^p$-norm $\left| \left( x_{1},\ldots ,x_{n}\right) \right| _{n,p}= \left( \sum \nolimits _{k=1}^{n}\left| x_{k}\right| ^{p}\right) ^{1/p}$. Hence,

$$\begin{aligned} E\left| X_{1}\right| ^{p}=\int \nolimits _{-1}^{1}\frac{\left| x_{1}\right| ^{p}}{2}\mathrm{d}x_{1}=2\int \nolimits _{0}^{1}\frac{x_{1}^{p}}{2} \mathrm{d}x_{1}=\left. \frac{x_{1}^{p+1}}{p+1}\right| _{0}^{1}=\frac{1}{p+1}. \end{aligned}$$

Moreover, for $n > 1$,

$$\begin{aligned} E\left| X_{n}\right| ^{p}= & {} \int \nolimits _{K_{n}}\left| x_{n}\right| ^{p}\cdot f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n}\\= & {} \int \nolimits _{K_{n-1}} \left( 2\int \nolimits _{0}^{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} x_{n}^{p}\cdot f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \mathrm{d}x_{n}\right) \\&\times \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \int \nolimits _{K_{n-1}} \left. \frac{x_{n}^{p+1}}{p+1} \right| _{0}^{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} \\&\times \frac{f_{n-1}\left( x_{1},x_{2},\ldots ,x_{n-1}\right) }{\left[ 1-\left( \left| x_{1}\right| ^{p} +\cdots + \left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \int \nolimits _{K_{n-1}}\frac{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] }{p+1} \cdot f_{n-1}\left( x_{1},x_{2},\ldots ,x_{n-1}\right) \\&\times \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \frac{1}{p+1}\left\{ 1-\left( E\left| X_{1}\right| ^{p}+E\left| X_{2}\right| ^{p}+\cdots +\left| X_{n-1}\right| ^{p}\right) \right\} . \end{aligned}$$

Thus,

$$\begin{aligned} E\left| X_{2}\right| ^{p}=\frac{1}{p+1}\left\{ 1-E\left| X_{1}\right| ^{p}\right\} =\frac{1}{p+1}\left( 1-\frac{1}{p+1}\right) = \frac{1}{p+1}\frac{p}{p+1}, \end{aligned}$$

and, by induction,

$$\begin{aligned} E\left| X_{n}\right| ^{p}= & {} \frac{1}{p+1}\left\{ 1-\left( \frac{1}{p+1} +\frac{p}{\left( p+1\right) ^{2}}+\cdots +\frac{p^{n-2}}{\left( p+1\right) ^{n-1} }\right) \right\} \\= & {} \frac{1}{\left( p+1\right) }\left\{ 1-\frac{1}{p+1}\cdot \frac{1-p^{n-1}/\left( p+1\right) ^{n-1}}{1-p/\left( p+1\right) }\right\} =\frac{p^{n-1}}{\left( p+1\right) ^{n}}. \end{aligned}$$

Therefore,

$$\begin{aligned} E\left( \sum \limits _{~n=1}^{\infty }\left| X_{2n-1}\right| ^{p}\right) =\sum \limits _{n=1}^{\infty }\frac{p^{2n-1-1}}{\left( p+1\right) ^{2n-1}}=\frac{1}{p+1}\cdot \frac{1}{1-p^{2}/\left( p+1\right) ^{2}}= \frac{p+1}{2p+1}, \end{aligned}$$

and

$$\begin{aligned} E\left( \sum \limits _{~n=1}^{\infty }\left| X_{2n}\right| ^{p}\right) =\sum \limits _{n=1}^{\infty }\frac{p^{2n-1}}{\left( p+1\right) ^{2n}}=\frac{p }{\left( p+1\right) ^{2}}\cdot \frac{1}{1-p^{2}/\left( p+1\right) ^{2}}= \frac{p}{2p+1}. \end{aligned}$$

In consequence,

$$\begin{aligned} E\left| \pi _{\mathrm{odd}}\left( \xi \right) \right| _{p}^{p}=\frac{ p+1}{2p+1},\quad E\left| \xi -\pi _\mathrm{odd}\left( \xi \right) \right| _{p}^{p}=\frac{p}{2p+1}, \end{aligned}$$

so that

$$\begin{aligned} E\left| \xi \right| _{p}^{p}=E\left( \sum \limits _{~n=1}^{\infty }\left| X_{n}\right| ^{p}\right) =\frac{p+1}{2p+1}+\frac{p}{2p+1}=1. \end{aligned}$$