 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/23f/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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 .

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