Measure, Integration & Real Analysis · Problem 3A.6

Integration — Sheldon Axler

Mathematics — Measure Theory · lower-lebesgue-sum-refinement

Problem

Suppose

(X, \mathcal{S}, \mu)

is a measure space,

f: X \to [0, \infty]

\mathcal{S}

-measurable, and

P

and

P'

are

\mathcal{S}

-partitions of

X

such that each set in

P'

is contained in some set in

P

. Prove that

\mathcal{L}(f, P) \leq \mathcal{L}(f, P')