Measure, Integration & Real Analysis · Problem 3A.7

Integration — Sheldon Axler

Mathematics — Measure Theory · weighted-counting-measure-integral

Problem

Suppose

X

is a set,

\mathcal{S}

is the

\sigma

-algebra of all subsets of

X

, and

w: X \to [0, \infty]

is a function. Define a measure

\mu

(X, \mathcal{S})

\mu(E) = \sum_{x \in E} w(x)

for

E \subseteq X

. Prove that if

f: X \to [0, \infty]

is a function, then

\int f\, d\mu = \sum_{x \in X} w(x) f(x),

where the infinite sums above are defined as the supremum of all sums over finite subsets of

E

(first sum) or

X

(second sum).