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

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-limit-exchange-counterexample

Problem

Give an example of a sequence of continuous real-valued functions

f_1, f_2, \ldots

[0, 1]

and a continuous real-valued function

f

[0, 1]

such that

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

for each

x \in [0, 1]

but

\int_0^1 f \neq \lim_{k \to \infty} \int_0^1 f_k.