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

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-constant-function

Problem

Suppose

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

is a bounded function such that

L(f, P, [a, b]) = U(f, P, [a, b])

for some partition

P

[a, b]

. Prove that

f

is a constant function on

[a, b]