Measure, Integration & Real Analysis · Problem 1B.4

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-sums-strict-subadditivity-example

Problem

Give an example of bounded functions

f, g : [0, 1] \to \mathbf{R}

such that

L(f, [0, 1]) + L(g, [0, 1]) < L(f + g, [0, 1])

and

U(f + g, [0, 1]) < U(f, [0, 1]) + U(g, [0, 1]).