 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008/d/d3/b/solution

S_{N} = \sum_{k = 0}^{N} a_{k}

are measurable, nonnegative and increasing. By the monotone convergence theorem,

\int k = 0 \sum \infty a_{k} = k = 0 \sum \infty \int a_{k} < \infty .

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.

 New to topics? Read the docs here!