# Characterization of $\mu$-measurable $\overline{\mathbb{R}}^+$-valued functions

First are relevant definitions from textbook $$\textbf{Analysis III}$$ by Amann.

Let $$(X, \mathcal{A}, \mu)$$ be a complete, $$\sigma$$-finite measure space and $$(E,|\cdot|)$$ a Banach space.

We say $$f \in E^{X}$$ is $$\mu$$-simple if $$f(X)$$ is finite, $$f^{-1}(e) \in \mathcal{A}$$ for every $$e \in E,$$ and $$\mu\left(f^{-1}(E \backslash\{0\})\right)<\infty .$$ We denote by $$\mathcal{S}(X, \mu, E)$$ the set of all $$\mu$$-simple functions.

A function $$f \in E^{X}$$ is said to be $$\mu$$-measurable if there is a sequence $$\left(f_{j}\right)$$ in $$\mathcal{S}(X, \mu, E)$$ such that $$f_{j} \rightarrow f$$ $$\mu$$-almost everywhere as $$j \rightarrow \infty$$. We set $$\mathcal{L}_{0}(X, \mu, E):=\left\{f \in E^{X} \mid f \text { is } \mu \text {-measurable}\right\}$$

In the theory of integration, it is useful to consider not only real-valued functions but also maps into the extended number line $$\overline{\mathbb{R}}$$. Such maps are called $$\overline{\mathbb{R}}$$-valued functions. An $$\overline{\mathbb{R}}$$-valued function $$f: X \rightarrow \overline{\mathbb{R}}$$ is said to be $$\mu$$-measurable if $$\mathcal{A}$$ contains $$f^{-1}(-\infty), f^{-1}(\infty)$$, and $$f^{-1}(O)$$ for every open subset $$O$$ of $$\mathbb{R} .$$ We denote the set of all $$\mu$$-measurable $$\overline{\mathbb{R}}$$-valued functions on $$X$$ by $$\mathcal{L}_{0}(X, \mu, \overline{\mathbb{R}})$$

The authors continue to present propositions:

My questions:

From the authors' definitions, we have

Def: $$f \in \mathcal{L}_{0} (X, \mu, E)$$ if and only if there is a sequence $$\left(f_{j}\right)$$ in $$\mathcal{S}(X, \mu, E)$$ such that $$f_{j} \rightarrow f$$ $$\mu$$-almost everywhere as $$j \rightarrow \infty$$.

and

Thm: $$f \in \mathcal{L}_{0} (X, \mu, \overline{\mathbb{R}}^{+})$$ if and only of there is an increasing sequence $$\left(f_{j}\right)$$ in $$\mathcal{S}\left(X, \mu, \mathbb{R}^{+}\right)$$ such that $$f_{j} \to f$$ for $$j \to\infty$$.

In light of Def, I fell it's very intuitive for the following statement (St) to be correct.

St: $$f \in \mathcal{L}_{0} (X, \mu, \overline{\mathbb{R}}^{+})$$ if and only of there is a sequence $$\left(f_{j}\right)$$ in $$\mathcal{S}\left(X, \mu, \mathbb{R}^{+}\right)$$ such that $$f_{j} \to f$$ $$\mu$$-almost everywhere for $$j \to\infty$$.

Clearly, Thm $$\implies$$ St. As such, I would like to ask if St $$\implies$$ Thm. If it's not the case, could you please give an intuitive explanation why it's not correct that St $$\implies$$ Thm?

Thank you so much for your clarification!

I've figured a proof and posted it as an answer here to peacefully close this question.

$$\textbf{Theorem}$$ If there is a sequence $$\left(f_{n}\right)$$ of $$\mu$$-simple functions $$f_n: X \to \mathbb{R}^{+}$$ that converges to $$f: X \to \overline{\mathbb{R}}^{+}$$ $$\mu$$-almost everywhere, then there is an increasing sequence $$\left(g_{m}\right)$$ of $$\mu$$-simple functions $$g_m: X \to \mathbb{R}^{+}$$ that converges to $$f: X \to \overline{\mathbb{R}}^{+}$$.

$$\textbf{My attempt}$$

First, we need the following lemma.

If there is a sequence $$\left(f_{n}\right)$$ of $$\mu$$-simple functions $$f_n: X \to \mathbb{R}^{+}$$ that converges to $$f: X \to \overline{\mathbb{R}}^{+}$$ $$\mu$$-almost everywhere, then $$\mathcal{A}$$ contains $$f^{-1}(+\infty)$$ and $$f^{-1}(O)$$ for every open subset $$O$$ of $$\mathbb{R}$$.

For all $$(m, k) \in \mathbb{N} \times\left\{0, \ldots, 2^{m}-1\right\}$$, we define $$\begin{cases} b_{m,k} &=k m 2^{-m} \\ B_{m,k} &=f^{-1} ([k m 2^{-m}, (k+1)m 2^{-m})) \\ B_m &= f^{-1} ([m, +\infty)) \end{cases}$$

It follows from our Lemma that $$B_{m,k}$$ and $$B_{m}$$ belong to $$\mathcal A$$. Then we define a sequence $$(g_m)$$ by $$g_{m}=\mathbb{1}_{B_{m}} + \sum_{k=0}^{2^{m}-1} b_{m,k} \mathbb{1}_{B_{m,k}}$$

It's not hard to verify that $$(g_m)$$ is an increasing sequence of $$\mu$$-simple functions and converges to $$f$$.