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

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 !

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