Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 23F / a

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- Solution a

Solution

For every

t \geq 0

, the fundamental theorem of calculus gives

φ (t) = \int_{0}^{t} φ^{'} (s) d s = \int_{0}^{\infty} φ^{'} (s) 1_{{s < t}} d s .

Apply this with

t = ∣ F (x) ∣

. Since the integrand is nonnegative, Tonelli theorem permits exchanging the integrals:

\int_{X} φ (∣ F (x) ∣) d μ (x) = \int_{X} \int_{0}^{\infty} φ^{'} (s) 1_{{s < ∣ F (x) ∣}} d s d μ (x) = \int_{0}^{\infty} φ^{'} (s) μ ({∣ F ∣ > s}) d s .

This is the generalized layer-cake representation.

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