Past exam of the options II course of the University of Oxford / 2008 / D / D3 / b / Solution

The partial sums $S_N={\sum }_{k=0}^N{a}_k$ are measurable, nonnegative and increasing. By the monotone convergence theorem,A nonnegative measurable function with finite integral is finite almost everywhere: if it were infinite on a positive-measure set, every constant multiple of that set's indicator would be a lower bound, forcing an infinite integral. Thus the series converges almost everywhere.