= Solution
The <monotone convergence theorem> states that if $f_n$ are nonnegative measurable functions with $f_n(x)\uparrow f(x)$ almost everywhere, then
$$
\int f_n\,d\mu\uparrow\int f\,d\mu,
$$
where either side may be infinite.
The integrals increase and are bounded above by $\int f$, so let their limit be $L\leq\int f$. Let $s$ be a nonnegative simple function with $s\leq f$, and fix $0<c<1$. The sets
$$
E_n=\{x:f_n(x)\geq c s(x)\}
$$
increase and cover $\{s>0\}$ up to a null set. Hence continuity of measure from below gives
$$
\int f_n\,d\mu
\geq c\int_{E_n}s\,d\mu
\longrightarrow c\int s\,d\mu.
$$
\b[Thus $L\geq c\int s$. Taking the supremum over all simple $s\leq f$ and then letting $c\uparrow1$ gives $L\geq\int f$. Therefore equality holds.]
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