Measure, Integration & Real Analysis · Problem 1A.14

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-uniform-convergence

Problem

Suppose

f_1, f_2, \ldots

is a sequence of Riemann integrable functions on

[a, b]

such that

f_1, f_2, \ldots

converges uniformly on

[a, b]

to a function

f : [a, b] \to \mathbf{R}

. Prove that

f

is Riemann integrable and

\int_a^b f = \lim_{n \to \infty} \int_a^b f_n.