Measure, Integration & Real Analysis · Problem 5B.2

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · iterated-integrals-double-sums

Problem

(a) Give an example of a doubly indexed collection

\{x_{m,n} : m, n \in \mathbf{Z}^+\}

of real numbers such that

\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} x_{m,n} = 0 \quad \text{and} \quad \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} x_{m,n} = \infty.

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