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The indefinite integral of a nonnegative simple function is a measure

Statement

Let $s$ be a nonnegative simple measurable function on $(X, A, μ)$ and define $ν_{s} (A) : = \int_{A} s d μ (A \in A) .$ Then $ν_{s}$ is a measure on $(X, A)$ .

Facts & Assumptions

Given: A nonnegative simple measurable function $s$ on $(X, A, μ)$ .

[L1]

For measurable $A$ , the set function $A \mapsto \int_{A} s d μ$ is defined as $A \mapsto \int s χ_{A} d μ$ (Integral over a measurable subset).

[L2]

The simple integral is additive and homogeneous on nonnegative simple functions (The simple integral is monotone, homogeneous, and additive).

[L4]

The integral is independent of the chosen finite measurable representation, so a disjoint partition including the zero-valued complement may be used (The simple integral is independent of the chosen representation).

[L3]

A measure is a set function with value $0$ at the empty set and countable additivity on pairwise disjoint measurable families (Measures on sigma-algebras).

Proof

technique · direct

1.1L1L2L4givenalgebra

By [L4], choose a finite measurable partition $X = ⨆_{j = 0}^{m} E_{j}$ on which $s = c_{j} \geq 0$ , including its zero-valued complement. For every measurable $A$ , the sets $A \cap E_{j}$ partition $A$ , so [L1] and [L2] give $ν_{s} (A) = \sum_{j = 0}^{m} c_{j} μ (A \cap E_{j})$ . A term with $c_{j} = 0$ is defined to be zero even when $μ (A \cap E_{j}) = + \infty$ .

2.1step 1.1L2L3algebra

Step 1.1 gives $ν_{s} (\emptyset) = 0$ . If $(A_{n})$ is pairwise disjoint, then for each fixed $j$ , the sets $(A_{n} \cap E_{j})$ are pairwise disjoint. Countable additivity of $μ$ and interchange of one finite sum with a nonnegative series give $ν_{s} (⋃_{n} A_{n}) = \sum_{j = 0}^{m} c_{j} \sum_{n} μ (A_{n} \cap E_{j}) = \sum_{n} \sum_{j = 0}^{m} c_{j} μ (A_{n} \cap E_{j}) = \sum_{n} ν_{s} (A_{n})$ . Zero-coefficient terms remain zero by the simple-integral convention.

3.1step 2.1L3∎

Therefore $ν_{s}$ satisfies the two conditions in [L3], so it is a measure.

Depends on

Used by

A nonnegative integral over a null set vanishes Corollary
Monotone convergence for the integral Theorem
The indefinite integral of a nonnegative measurable function is a measure Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard F. Bass, Real Analysis for Graduate Students, ch. 7 (standard reference, not scraped)
John K. Hunter, Measure Theory Notes, §4.2 (standard reference, not scraped)