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

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-sums-subadditivity

Problem

Suppose

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

are bounded functions. Prove that

L(f, [a, b]) + L(g, [a, b]) \leq L(f + g, [a, b])

and

U(f + g, [a, b]) \leq U(f, [a, b]) + U(g, [a, b]).