Exploration of indispensable Banach-space valued functions

1.   Introduction

The property of $L^p(\Omega, \mathscr{F}, \mu; \mathcal{B})$ will be discussed, here $(\Omega, \mathscr{F}, \mu)$ is a $\sigma$-finite measure space, and $\mathcal{B}$ is a real Banach space. For $1\le p < \infty$, $L^p(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is a linear space of all $\mathcal{B}$-valued Bochner $L^p$ integral function with the norm given by the formula

If $p = \infty$, $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is a linear space of all $\mathcal{B}$-valued essential bounded function with norm defined by letting

If $\mathcal{B} = \mathbb{R}$, when $p\in[1, \infty]$, it is known that there exists a sequence of finite-valued simple measurable function $\left\{F_n, n\ge1\right\}$ such that

for any $F\in L^p(\Omega, \mathscr{F}, \mu; \mathbb{R})$ (see ^[1,2]). If $\mathcal{B}$ is a general Banach space, $p\in[1, \infty)$, there exists a sequence of finite-valued simple measurable function $F_n$ such that

for any $F\in L^p(\Omega, \mathscr{F}, \mu; \mathcal{B})$, and there exists a sequence of countable-valued simple measurable function $F_n$ such that

for any $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ (see^[3]).

The difference between infinite dimension Banach space $\mathcal{B}$ and $\mathbb{R}$ is that the closed ball of $\mathbb{R}$ is compact set and the closed ball of $\mathcal{B}$ is non-compact set (see^[4]), which makes the property of $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is very different form $L^\infty(\Omega, \mathscr{F}, \mu; \mathbb{R})$.

Convergence methods of Banach-valued function were defined in serval ways. For example, Zheng and Cui^[5] investigated that $l^\infty(X)$- evaluation uniform convergence of operator series can be described completed by the essential bounded subset of $l^\infty(X)$. Here $X$ is a Banach space,

and $l^\infty(X)$ equip the norm of

León-Saavedra considered unconditionally convergence of a series $\sum_ix_i$ in a Banach space. ^[6] showed that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. Furthermore, this paper unifies several versions of the Orlicz-Pettis theorem that incorporate summability methods. ^[7] give a another version of the Orlicz-Pettis theorem within the frame of the strong $\rho$-Cesàro convergence. ^[8] unified several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated with the wuc series. ^[9] gave a new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong $\rho$-Cesàro summability is obtained.

In this work, we will present a necessary and sufficient condition for the existence of $F_n$ in $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ for $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$, by constructing a sequence of finite-valued measurable functions that converge to $F$ in some sense. A counterexample is also discussed to demonstrate that there exists $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ for which $F_n$ cannot converge to $F$ in the norm topology of $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ for any sequence of finite-valued measurable functions $F_n$.

2.   Preliminaries

The following definitions are about Banach-valued measurable function.

Definition 2.1. ^[10] If $(\Omega, \mathscr{F})$ is a measurable space, $\mathcal{B}$ is a Banach space, $\Omega_1, \cdots, \Omega_n\in\mathscr{F}$ are pairwise disjoined nonempty sets, $x_1, \cdots, x_n\in\mathcal{B}$, then the map

is called finite-valued simple function. And the map

is called countable-valued simple function. A map $F:\Omega\to\mathcal{B}$ is called measurable if $\forall A\in\mathscr{B}(\mathcal{B}), F^{-1}(B)\in\mathscr{F}$. $F$ is called strongly measurable if there is a sequence of finite-valued simple function $F_n$ such that $\forall\omega\in\Omega$,

Definition 2.2. ^[11] Let $F:\Omega\to\mathcal{B}$ be a map, for all $f\in\mathcal{B}^*$, the function $f(F(\omega))$ is measurable function on $(\Omega, \mathscr{F}, \mu)$, then $F$ is called weak measurable function on $(\Omega, \mathscr{F}, \mu)$.

The following theorem describes the relationship weak and strong measurable.

Theorem 2.1. (Pettis)^[11] Let $F:\Omega\to\mathcal{B}$ be a map, the following assertions are equivalent:

(1) $F$ is strongly measurable.

(2) $F$ is weakly measurable and $F(\Omega)$ is almost separable.

By Theorem 2.1, if $\mathcal{B}$ is separable space, then $F$ is strongly measurable if and only if it's weakly measurable.

Then the definition of Bochner $L^P$-space is given as follows.

Definition 2.3. ^[3,10] Let $(\Omega, \mathscr{F}, \mu)$ be a measure space, and let $F:\Omega\to\mathcal{B}$ be a finite-valued simple function with a form of

If $\sum_{i = 1}^n\mu(\Omega_i) < \infty$, then the Bochner integral of $F$ is defined by

And let $F:\Omega\to\mathcal{B}$ be a strongly measurable function. If there exists a $p\in[1, \infty)$ such that

then $F$ is called $L^p$-integrable on $(\Omega, \mathscr{F}, \mu)$. The linear space of all $L^p$-integrable function with the following seminorm

is denoted by $L^p\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$. If the function

is essential bounded, then $F$ is called essential bounded. The linear space of all essential bounded function with the following seminorm

is denoted by $L^\infty\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$.

The following theorems show that the collection of finite-valued function is dense in $L^p\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$ if $p\in[1, \infty)$, and the collection of countable-valued function is dense in $L^\infty\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$.

Theorem 2.2. ^[3] Let $(\Omega, \mathscr{F}, \mu)$ be a measurable space, $F:\Omega\to\mathcal{B}$ is a strongly measurable function, $p\in[1, \infty)$, then the following statements are the same in meaning:

(1) $F\in L^p\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$.

(2) There exists a sequence of finite-valued simple function $F_n$ such that

Theorem 2.3. ^[3] Let $(\Omega, \mathscr{F}, \mu)$ be a measurable space, $F:\Omega\to\mathcal{B}$ be a strongly measurable function, then the following statements are synonymous:

(1) $F\in L^\infty\left(\Omega, \mathscr{F}, \mu; \mathcal{B}\right)$.

(2) There exists a sequence of countable-valued simple function $F_n$ such that

3.   Main result

Theorem 3.1. If $(\Omega, \mathscr{F}, \mu)$ is a measure space, and $\mathcal{B}$ is a real Banach space, $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$, then the following assertions are equivalent:

(1) There exists a sequence of finite-valued simple function $F_n$ such that

(2) There exists a measurable set $\tilde{\Omega}\in\mathscr{F}$ such that $\mu(\tilde{\Omega}) = 0$ and $F(\tilde{\Omega}^c)$ is a sequential compact set.

Proof. If (1) holds, suppose

and

where $\left\{E_{in}\right\}_{i = 1}^{K_n}$ are pairwise disjoined and $\cup_{i = 1}^{K_n}E_{in} = \Omega$. By the definition of essential bounded, there exists $\tilde{E}_n\in\mathscr{F}$ such that $\mu(\tilde{E}_n) = 0$ and

Considering

Then $\mu(\tilde{\Omega}) = 0$. Let

then there exist $i = 1, \cdots, K_n$ such that

Therefore, $\left\{x_{in}\right\}_{i = 1}^{K_n}$ is a $1/n-$ web of $F(\tilde{\Omega}^c)$. By the arbitrary of $n$, $F(\tilde{\Omega}^c)$ is a sequential compact set.

If condition (2) is satisfied, then $\forall n\in\mathbb{N}_+$, there exists a finite $1/n-$ web $\left\{x_{in}\right\}_{i = 1}^{K_n}$ of $F(\tilde{\Omega}^c)$. Let

Let $\tilde{E}_{1n} = E_{1n}$, and for $i > 1$, defined by

Now, let's define a finite-valued function

then

By the arbitrary of $n$, (1) holds. □

From now on, suppose $\mathcal{B}$ is real Banach space which dual space $\mathcal{B}^*$ is separable, and $(\Omega, \mathscr{F}, \mu)$ is complete measure space. Then $\mathcal{B}$ is separable. Let $\left\{f_n\right\}_{n\in\mathbb{N}_+}$ be countably dense subset of $\mathcal{B}^*$. We define a new convergence.

Definition 3.1. Let $F_k, k = 1, 2, \cdots$ be a sequence of $\mathcal{B}$-valued strongly measurable function on $(\Omega, \mathscr{F}, \mu)$, we say $F_k$ weakly converge to a $\mathcal{B}$-valued function $F$ almost uniformly if there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$ and for all weak neighborhood of origin $W$, there exists $N\in\mathbb{N}_+$ such that $\forall k > N$,

We say $F_i$ is a almost uniformly weak Cauchy sequence if there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$ and for all weak neighborhood of origin $W$, there exists $N\in\mathbb{N}_+$ such that $\forall i, j > N$,

Theorem 3.2. (1) $F_k$ weakly converge to $F$ almost uniformly if and only if there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and for all $f\in\mathcal{B}^*$, then

(2) $F_k$ is a almost uniformly weak Cauchy sequence if and only if there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and for all $f\in\mathcal{B}^*$, then

Proof. We have just proven (1), and likewise, (2) can be demonstrated. Suppose there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and for all weak neighborhood of origin $W$, there exists $N\in\mathbb{N}_+$ such that $\forall k > N$,

Let $f\in\mathcal{B}^*$, given $m\in\mathbb{N}_+$, consider the set

Then, for $N\in\mathbb{N}_+$ such that $\forall k > N$,

That is

Let $k\to\infty$,

By the arbitrary of $m$,

Suppose there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and for all $f\in\mathcal{B}^*$, we have

Given a weak neighborhood of origin $W$, by the definition of weak topology, there exists $g_1, \cdots, g_m\in\mathcal{B}^*$ and $\epsilon > 0$ such that

Then, for $i = 1, \cdots, m, \exists N_i\in\mathbb{N}_+$ such that

Let $N = max(N_1, \cdots, N_m)$, then $\forall k > N$, we have

Therefore, $F_k$ weakly converge to $F$ almost uniformly. □

Theorem 3.3. Let $F_k, k = 1, 2, \cdots$ be a sequence of $\mathcal{B}$-valued strongly measurable function. If $F_k$ weakly converge to $F$ almost uniformly, then $F$ is strongly measurable.

Proof. By Theorem 2.1, it is sufficient to prove that $F$ is weakly measurable. If $F_k$ weakly converge to $F$ almost uniformly, then there exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and $\forall f\in\mathcal{B}^*$,

Therefore, $f(F_k\mathbb{I}_{E^c})$ pointwise converge to $f(F\mathbb{I}_{E^c})$. By the arbitrary of $f$, $F\mathbb{I}_{E^c}$ is weakly measurable, thus it is strongly measurable. Because $\mu(E) = 0$ and $(\Omega, \mathscr{F}, \mu)$ is complete, $F\mathbb{I}_E$ is strongly measurable. In summary, $F = F\mathbb{I}_{E^c}+F\mathbb{I}_E$ is measurable. □

Theorem 3.4. If $F_k$ weakly converge to $F'$ and $F''$ almost uniformly, then $F' = F'', \mu-a.e.$.

Proof. If $F_k$ weakly converge to $F'$ and $F''$ almost uniformly, then there exist $E', E''\in\mathscr{F}$ such that $\mu(E') = \mu(E'') = 0$, and

Then

Therefore, $F' = F'', \mu-a.e.$. □

Theorem 3.5. Let $F, F_k, k = 1, 2, \cdots\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$, then $F_k$ weakly converge to $F$ almost uniformly if and only if

(1) $\sup_{k\in\mathbb{N}_+}\left\Arrowvert F_k\right\Arrowvert_{L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})} < \infty$.

(2) There exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and $\forall n\in\mathbb{N}_+$,

Proof. Suppose $F_k$ weakly converge to $F$ almost uniformly, since (2) is self-evident, we will focus on demonstrating (1). By the conditions, there exists a $E\in\mathscr{F}$ such that $\mu(E) = 0$ and

In addition, we can suppose $\sup_{\omega\in E^c}\left\Arrowvert F(\omega)\right\Arrowvert < \infty$ and $\sup_{\omega\in E^c}\left\Arrowvert F_k(\omega)\right\Arrowvert < \infty(k\in\mathbb{N}_+)$. Fixed $f\in\mathcal{B}^*$, then there exists $k_0\in\mathbb{N}_+$ such that for $k\ge k_0$,

For $k\ge k_0$,

Therefore,

By Uniform Boundedness Principle,

Thus, $\sup_{k\in\mathbb{N}_+}\left\Arrowvert F_k\right\Arrowvert_{L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})} < \infty$. Now we suppose (1) and (2) are true, then there exists a $E\in\mathscr{F}$ such that $\mu(E) = 0$ and $\forall n\in\mathbb{N}_+$,

We can assume

Fixed $f\in\mathcal{B}^*$, then $\forall\epsilon > 0, \exists n_0\in\mathbb{N}_+$ such that

Then $\forall k\in\mathbb{N}_+, \forall\omega\in E^c$,

By the arbitrary of $\omega$,

Therefore,

By the arbitrary of $\epsilon$,

Finally, by the arbitrary of $f$, $F_k$ weakly converge to $F$ almost uniformly □

Theorem 3.6. Suppose $F$ is a $\mathcal{B}$-valued strongly measurable function on $(\Omega, \mathscr{F}, \mu)$, then $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ if and only if there exists a sequence of finite-valued simple function $F_k$ such that $F_k$ weakly converge to $F$ almost uniformly.

Proof. Suppose $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$, then there is a $E\in\mathscr{F}$ such that $\mu(E) = 0$, and

By Banach-Alaoglu theorem, $F(E^c)$ is weak relatively compact sets. Let $k\in\mathbb{N}_+$, and

Then there exist $\forall k\in\mathbb{N}_+, \exists\left\{x_{ik}\right\}_{i = 1}^{N_k}\subset F(E^c)$ such that

Let

and $\tilde{E}_{1k} = E_{1k}$, for $i > 1$, we can define

We can construct a finite-valued measurable function

Because $F_k(E^c)\subset F(E^c)$,

For all $n\in\mathbb{N}_+$, for $k > n$,

Therefore,

By Theorem 3.5, $F_k$ weakly converge to $F$ almost uniformly.

Consider the possibility that a sequence of finite-valued simple function $F_k$ that weakly converge to $F$ almost uniformly, then there exists a $E\in\mathscr{F}$ such that $\mu(E) = 0$ and $\forall f\in\mathcal{B}^*$,

Then there exists a $k_0\in\mathbb{N}_+$ such that

Because $F_{k_0}$ is finite-valued function,

By Uniform Boundedness Principle,

Therefore, $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$. □

Now we will proof $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is complete in the sense of weak convergence almost uniformly.

Theorem 3.7. Let $F_k, k = 1, 2, \cdots\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ be a almost uniformly weak Cauchy sequence, then $\exists F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ such that $F_k$ weakly converge to $F$ almost uniformly.

Proof. Suppose $F_k\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is a almost uniformly weak Cauchy sequence, then there exists a $E\in\mathscr{F}$ such that $\mu(E) = 0$ and for all $k\in\mathbb{N}_+$,

And $\forall f\in\mathcal{B}^*$,

Fixed $f\in\mathcal{B}^*, \exists k_0\in\mathbb{N}_+$ such that $\forall k > k_0$,

For $i = 1, \cdots, k_0$,

Therefore,

By Uniform Boundedness Principle, there exists $M\in(0, \infty)$ such that

Fixed $\omega\in E^c$, $\left\{F_k(\omega)\right\}_{k\in\mathbb{N}_+}$ is a bounded sequence, which means it's a weak relatively compact sequence. Therefore exists a subsequence $\left\{F_{k_i}(\omega)\right\}_{i\in\mathbb{N}_+}$ and $F(\omega)\in\mathcal{B}$ such that $F_{k_i}(\omega)$ weakly converge to $F(\omega)$. Because $\left\{F_k(\omega)\right\}_{k\in\mathbb{N}_+}$ is a weak cauchy sequence, $\forall f\in\mathcal{B}^*, \forall\epsilon > 0, \exists K\in\mathbb{N}_+$ such that $\forall m, n\ge K$,

Meanwhile, $\exists i_0\in\mathbb{N}_+$ such that $k_{i_0} > N$ and

when $k > k_{i_0}$,

which means $F_k(\omega)$ weakly converge to $F(\omega)$. By Mazur Theorem,

where $\bar{co}\left\{F_k(\omega)\right\}_{k\in\mathbb{N}_+}$ is convex hull of $\left\{F_k(\omega)\right\}_{k\in\mathbb{N}_+}$ in norm topology of $\mathcal{B}$. Therefore, $F$ is essential bounded, and

Finally, $F_k$ weakly converge to $F$ almost uniformly can be show. Fixed $f\in\mathcal{B}^*$, there exists a $\left\{F_{k_i}\right\}_{i\in\mathbb{N}}$ such that

Then

Let $j\to\infty$,

$\forall\epsilon > 0, \exists K\in\mathbb{N}_+$ such that $\forall m, n\ge K$,

Meanwhile, $\exists i_0\in\mathbb{N}_+$ such that $k_{i_0} > N$ and

Therefore, for $k > k_{i_0}$, we have

which means $F_k$ weakly converge to $F$ almost uniformly. □

Finally, we will provide a counterexample to demonstrate that there exists an $F \in L^\infty(\Omega; \mathcal{B})$ for which $F_n$ cannot converge to $F$ in the norm topology of $L^\infty(\Omega; \mathcal{B})$ for any sequence of finite-valued measurable functions $F_n$.

Let $\mathcal{H} = L^2([-\pi, \pi], \mathscr{B}([-\pi, \pi]), l)$, where $l$ is Lebesgue measure. It is obvious that $\mathcal{H}$ is real separable Hilbert space and $\mathcal{H}' = \mathcal{H}$ by Riesz Representation Theorem, which means the dual space of $\mathcal{H}$ is separable.

Let $n\in\mathbb{N}$,

Then $\left\{h_n\right\}_{n\in\mathbb{N}}$ is the unit orthogonal basis of $\mathcal{H}$, and

A measure space $([0, 1], \mathscr{B}([0, 1]), l)$ is given. Let $\mathcal{C}$ be the Cantor set of $[0, 1]$, and the countable connected component of $\mathcal{C}^c$ denote by $\left\{E_n\right\}_{n\in\mathbb{N}_+}$.

Then $F\in L^\infty([0, 1], \mathscr{B}([0, 1]), l;\mathcal{H})$, and

However, given any zero measure set $E$, $F(E^c) = \left\{h_n\right\}_{n\in\mathbb{N}}$, which means $F(E^c)$ in not a sequential compact set. Therefore, any sequence of finite-valued measurable function cannot converge to $F$ in norm topology of $L^\infty([0, 1], \mathscr{B}([0, 1]), l;\mathcal{H})$. Let

Then $F_k$ are finite-valued measurable functions. Given $m\in\mathbb{N}_+$, for $k > m$,

and $l(\mathcal{C}) = 0$, which means $F_k$ weakly converge to $F$ almost uniformly.

4.   Prospect

In this paper, we give a necessary and sufficient condition for the existence of a sequence of finite-valued measurable function which converge to $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ in topology of essential supremum and give a new convergence which makes any $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ can find sequence of finite-valued measurable function which converge to $F$. On this basis, we can raise some valuable problems:

(1) Definition 3.1 and the proof of Theorem 3.2 to Theorem 3.6 depend on the separability of $\mathcal{B}^*$, if we can extended the conclusion to the condition that $\mathcal{B}^*$ is a general Banach space?

(2) If we can use a topology to convergence defined in Definition 3.1. For example, let $f\in\mathscr{B}^*$, we define the seminorm $p_f$ by

If we can prove that $p_f(F) = 0$ for all $f$ implies $\left\Arrowvert F\right\Arrowvert_{L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})} = 0$, then the the family $\mathscr{P}\equiv\left\{p_f:f\in\mathcal{B}^*\right\}$ can determine a new topology to characterize the convergence.

(3) Based on (2) and Theorem 3.2, we guess the following assertions are equivalent:

(a) $F_k$ weakly converge to $F$ almost uniformly.

(b) For all $f\in\mathcal{B}^*$, we have

(c) There exists $E\in\mathscr{F}$ such that $\mu(E) = 0$, and for all $f\in\mathcal{B}^*$, we have

Here $F, F_k, k = 1, 2, \cdots\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$.

5.   Conclusions

In the work, a necessary and sufficient condition for the existence of a sequence of finite-valued measurable function which converge to any given $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ is given. A new convergence is defined. In this convergence, any $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ has a sequence of finite-valued measurable function which converge to $F$. Finally, a counterexample is also given to show that there exists $F\in L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ for which $F_n$ cannot converge to $F$ in the norm topology of $L^\infty(\Omega, \mathscr{F}, \mu; \mathcal{B})$ for any sequence of finite-valued measurable functions $F_n$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 11761074), Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (No. 20200301053RQ).

Conflict of interest

The authors declare no conflict of interest.