How to prove this condition for Bochner integrability on a general measure space?

Solution 1:

Think of $A_n$ as a measure space with the restriction of the $\sigma-$ algebra on $\Omega$ and the restriction of the measure $\mu$. Since you already know the result for finite measure space you can find a simple function $t_n$ on this space such that $\int_{A_n} \|f\chi_{A_n}-t_n\|d\mu <\frac 1n$. Let $s_n=t_n$ on $A_n$ and $0$ outside. Then $s_n $ is a simple function on $\Omega$ and $\int \|f-s_n\|d\mu \leq \int_{A_n} \|f\chi_{A_n}-t_n\|+\int_{\Omega \setminus A_n} \|f\|d\mu$. Now $\int_{\Omega \setminus A_n} \|f\|d\mu \to 0$ because $\lim_n \int_{A_n} \|f\|d\mu =\int \|f\chi_{\{f \neq 0\}}\|d\mu (\equiv \int \|f\|d\mu)$ by Monotone ConvegenceTheorem.