 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/d/d4/b/solution

s_{n} = \sum_{k = 1}^{n} a_{k}

are nonnegative measurable functions and increase to an extended-valued function

s

\int_{R} s (x) d x = lim_{n \to \infty} \sum_{k = 1}^{n} \int_{R} a_{k} (x) d x = \sum_{k = 1}^{\infty} \int_{R} a_{k} (x) d x < \infty.

+ \infty

on a set of positive measure: otherwise its integral would exceed

M

times that measure for every

M > 0

. Thus the series converges to a finite value almost everywhere. Redefine it to be zero on the exceptional null set if a finite-valued representative is desired. The resulting function is Lebesgue integrable, with

\int k = 1 \sum \infty a_{k} = k = 1 \sum \infty \int a_{k} .

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