Show that an $\mathcal{A}$-integrable function is $\mu$-integrable

Question

I am reading Measure Theory by Donald L. Cohn and practicing my understanding on exercices. I find this one challenging :

Let ($X,\mathcal{A},\mu$) be a measure space and let $f : X → \bar{\mathbb{R}}$ be an $\mathcal{A}$-measurable function. Suppose that nested sets $A_1$ ⊆ $A_2$ ⊆ ..., all in $\mathcal{A}$, satisfy $\cup_{n=1}^{\infty}A_n=X$ and $lim_{n→∞}\int_{A_{n}}|f|d\mu<∞$. Show that f is $\mu$-integrable.

Thank you for your help !

Answer 1

$\int_{A_n}|f|d\mu=\int_X 1_{A_n}|f|d\mu$.

Since $1_{A_n}|f|$ increases and pointwisely converges to $1_{\cup_{n=1}^{\infty}A_n}|f|=1_X |f|=|f|$,

$\int_X 1_{A_n}|f|d\mu$ increases and converges to $\int_X |f|d\mu$, by a standard theorem of measure theory. By the hypothesis this limit is finite, i.e. $f$ is $\mu$-integrable.