 Solution

The monotone convergence theorem states that if $0\le h_n\uparrow h$ pointwise for measurable functions, then $h$ is measurable and $\int h={\lim }_n\int h_n$, allowing infinity.

Apply it to the partial sums $S_N={\sum }_{n=0}^N{f}_n$. They are nonnegative, measurable, and increase pointwise to $f$. Finite additivity of the integral and monotone convergence yieldThe assumed finite sum of integrals makes $f$ integrable. This proves the series-integration rule needed below directly from the stated theorem, without assuming an additional interchange theorem.