Measure, Integration & Real Analysis · Problem 5B.1

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · iterated-integrals-tonelli-fubini

Problem

(a) Let

\lambda

denote Lebesgue measure on

[0, 1]

. Show that

\int_{[0,1]} \int_{[0,1]} \frac{x^2 - y^2}{(x^2 + y^2)^2} \, d\lambda(y) \, d\lambda(x) = \frac{\pi}{4}

and

\int_{[0,1]} \int_{[0,1]} \frac{x^2 - y^2}{(x^2 + y^2)^2} \, d\lambda(x) \, d\lambda(y) = -\frac{\pi}{4}.

(b) Explain why (a) violates neither Tonelli's Theorem nor Fubini's Theorem.