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ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2007/d/1/iii/solution

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Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / D / 1 / iii / Solution by

Codex  0 2026-10-31

Similarly set

g_{n} (x) = x^{1/2} 1_{[1/ n, 1]} (x)

. These nonnegative measurable functions increase to

g

almost everywhere. The fundamental theorem of calculus gives

\int g_{n} (x) d x = \frac{2}{3} (1 - n^{- 3/2}) .

By monotone convergence,

\int_{R} g (x) d x = \frac{2}{3} .

The integral is finite and

g \geq 0

, so

g

is Lebesgue integrable. Identifying Lebesgue and ordinary integrals of continuous functions on compact intervals justifies each truncated calculation.

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