Measure, Integration & Real Analysis · Problem 1A.6

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-finitely-many-exceptions

Problem

Suppose

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

is Riemann integrable. Suppose

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

is a function such that

g(x) = f(x)

for all except finitely many

x \in [a, b]

. Prove that

g

is Riemann integrable on

[a, b]

and

\int_a^b g = \int_a^b f.