Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / D / D4 / b

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- Solution b

Solution

The partial sums

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

are nonnegative measurable functions and increase pointwise to the extended-valued sum

S = \sum_{k = 1}^{\infty} a_{k}

. The monotone convergence theorem and finite linearity give

\int S d x = lim_{n} \sum_{k = 1}^{n} \int a_{k} d x = \sum_{k = 1}^{\infty} \int a_{k} d x < \infty.

A nonnegative measurable function with finite integral is finite almost everywhere; if it were infinite on a positive-measure set, its integral would be infinite. Define it arbitrarily on the null exceptional set if a real-valued representative is desired. Hence the series converges almost everywhere to an integrable function and

\int k = 1 \sum \infty a_{k} (x) d x = k = 1 \sum \infty \int a_{k} (x) d x .

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