Measure, Integration & Real Analysis · Problem 3A.8

Integration — Sheldon Axler

Mathematics — Measure Theory · pointwise-zero-integral-nonzero

Problem

Suppose

\lambda

denotes Lebesgue measure on

\mathbf{R}

. Give an example of a sequence

f_1, f_2, \ldots

of simple Borel measurable functions from

\mathbf{R}

[0, \infty)

such that

\lim_{k \to \infty} f_k(x) = 0

for every

x \in \mathbf{R}

but

\lim_{k \to \infty} \int f_k\, d\lambda = 1