Measure, Integration & Real Analysis · Problem 1A.3

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integrable-epsilon-criterion

Problem

Suppose

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

is a bounded function. Prove that

f

is Riemann integrable if and only if for each

\varepsilon > 0

, there exists a partition

P

[a, b]

such that

U(f, P, [a, b]) - L(f, P, [a, b]) < \varepsilon.