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The simple integral is independent of the chosen representation

Statement

If a nonnegative simple measurable function $s$ admits two representations $s = \sum_{i = 1}^{m} c_{i} χ_{E_{i}} = \sum_{j = 1}^{n} d_{j} χ_{F_{j}},$ then the two coefficient sums defining $\int s d μ$ are equal. So The integral of a nonnegative simple function is well defined.

Facts & Assumptions

Given: Two simple representations of the same nonnegative simple measurable function $s$ .

[L1]

The simple integral is defined by $\int s d μ = \sum c_{j} μ (E_{j})$ with the convention $0 \cdot (+ \infty) = 0$ (The integral of a nonnegative simple function).

[L2]

A measure is countably additive on pairwise disjoint measurable families, hence finitely additive on finite measurable partitions (Measures on sigma-algebras).

Proof

technique · direct

1.1

For each pair $(i, j)$ put $G_{i j} : = E_{i} \cap F_{j}$ . The family $(G_{i j})$ is [given, L2] measurable and pairwise disjoint, and $E_{i} = ⨆_{j} G_{i j}, F_{j} = ⨆_{i} G_{i j} .$ Whenever $G_{i j} \neq \emptyset$ , the two simple formulas for $s$ agree on $G_{i j}$ , so $c_{i} = d_{j}$ .

2.1

Finite additivity over the partitions in step 1.1 gives[step 1.1, L1, L2, algebra] $\sum_{i} c_{i} μ (E_{i}) = \sum_{i, j} c_{i} μ (G_{i j}) = \sum_{i, j} d_{j} μ (G_{i j}) = \sum_{j} d_{j} μ (F_{j}) .$ If some coefficient is $0$ on a cell of infinite measure, the convention in [L1] forces both corresponding terms to be $0$ , so no ambiguity occurs there either.

3.1

Therefore the simple integral does not depend on the chosen representation, [step 2.1, L1] ∎ and the definition in [L1] is well defined.

Depends on

Used by

The nonnegative integral agrees with the simple integral on simple functions Proposition
The simple integral is monotone, homogeneous, and additive Proposition

Cited to discharge well-definedness by The integral of a nonnegative simple function.

Dependency tree · two levels

6 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

John K. Hunter, Measure Theory Notes, Definition 4.1 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)