limit of countably valued functions measurable Let $(\Omega, F, \mu)$ be a finite measure space, $Y$ a banach space and $f \colon \Omega \to Y$ a function such that there exists a sequence $(f_n)_n$ of countably valued functions of the form
$$ f_n=\sum_{m=1}^\infty y_{n,m} 1_{A_{n,m}},$$
where $y_{n,m} \in Y$ and $A_{n,m} \in F$, converging $\mu$ almost everywhere to $f$.
I was told that it is now possible to clip of the $f_n$'s such that there is a sequence of simple functions converging almost everywhere to $f$. Any ideas on how to achieve this?
 A: Since $\mu$ is continuous from above, we can find for each $n \in \mathbb{N}$ some $K_n \in \mathbb{N}$ such that
$$\left| \mu \left( \bigcup_{m=1}^{K_n} A_{n,m} \right)- \mu \left( \bigcup_{m=1}^{\infty} A_{n,m} \right) \right| \leq 2^{-n}.$$
The function
$$g_n := \sum_{m=1}^{K_n} y_{n,m} 1_{A_{n,m}}$$
is a simple function and $\mu(f_n \neq g_n) \leq  2^{-n}$ for all $n \in \mathbb{N}$. As
$$\begin{align*} &\mu \left( \{g_n \, \text{does not converge to $f$}\} \right) \\ &\leq \mu \left( \{ g_n \, \text{does not converge to $f$}\} \cap \bigcap_{n =N}^{\infty} \{f_n = g_n\} \right) + \mu \left( \bigcup_{n =N}^{\infty} \{f_n \neq g_n\} \right) \end{align*}$$
for each $N \in \mathbb{N}$, we get
$$\begin{align*} \mu \left( \{g_n \, \text{does not converge to $f$}\} \right) &\leq \mu \left( \{ f_n \, \text{does not converge to $f$}\}\right) + \sum_{n =N}^{\infty} \mu(f_n \neq g_n) \\ &\leq 0 + \sum_{n =N}^{\infty} 2^{-n}. \end{align*}$$
Since $N$ is arbitrary, this proves that $g_n$ converges almost everywhere to $f$.
