Let $(f_n)$ be an increasing sequence of integrable function such that $(f_n) \rightarrow f$ on $X$ and $f$ is integrable.
Show that $\int_X f = \lim_n \int_X f_n$.
2. Since $f_n(x)\longrightarrow f(x)$ and $\vert f_n(x) \vert \leq \vert f(x) \vert$ and $f$ is integrable, then by the Dominated convergence theorem the result follows. This is valid if $X \subset \mathbb{R} ^n$ with Lebesgue measure. I don't know if the DCT holds for arbitrary measure spaces.