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The simple integral is monotone, homogeneous, and additive

Statement

Let $s, t$ be nonnegative simple measurable functions and let $c \geq 0$ .

If $s \leq t$ pointwise, then $\int s d μ \leq \int t d μ$ .
$\int c s d μ = c \int s d μ$ .
$\int (s + t) d μ = \int s d μ + \int t d μ$ .

Facts & Assumptions

Given: Nonnegative simple measurable functions $s, t$ and a scalar $c \geq 0$ .

[L1]

The simple integral is well defined, so any convenient common refinement of the chosen simple representations may be used to compute it (The simple integral is independent of the chosen representation).

[L2]

The simple integral of $\sum_{j} a_{j} χ_{E_{j}}$ is $\sum_{j} a_{j} μ (E_{j})$ with $0 \cdot (+ \infty) = 0$ (The integral of a nonnegative simple function).

Proof

technique · direct

1.1

Choose one finite measurable partition $(E_{r})$ on which both $s$ and $t$ [L1, construct] are constant, say $s = \sum_{r} a_{r} χ_{E_{r}}$ and $t = \sum_{r} b_{r} χ_{E_{r}}$ . Then $a_{r} \leq b_{r}$ for every $r$ because $s \leq t$ .

2.1

Using the common partition from step 1.1 and [L2], [step 1.1, L2, algebra] $\int s d μ = \sum_{r} a_{r} μ (E_{r}), \int t d μ = \sum_{r} b_{r} μ (E_{r}), \int c s d μ = \sum_{r} c a_{r} μ (E_{r}),$ and $\int (s + t) d μ = \sum_{r} (a_{r} + b_{r}) μ (E_{r}) .$ Termwise comparison gives monotonicity, while ordinary finite-sum algebra gives homogeneity and additivity.

3.1

Therefore all three properties hold for the simple integral.

step 2.1∎

Depends on

Used by

Additivity of the nonnegative Lebesgue integral Corollary
The nonnegative integral agrees with the simple integral on simple functions Proposition
Monotone convergence for the integral Theorem
The indefinite integral of a nonnegative simple function is a measure Theorem

Dependency tree · two levels

4 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, Proposition 4.3 (standard reference, not scraped)