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The nonnegative integral agrees with the simple integral on simple functions

Statement

If $s$ is a nonnegative simple measurable function, then its nonnegative Lebesgue integral equals its simple integral: $\int s d μ = \int_{s i m p l e} s d μ .$

Facts & Assumptions

Given: A nonnegative simple measurable function $s$ .

[L1]

The nonnegative integral is the supremum of the simple integrals of all simple minorants $0 \leq u \leq s$ (The nonnegative Lebesgue integral).

[L2]

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

[L3]

The simple integral itself is well defined on every nonnegative simple function (The integral of a nonnegative simple function, The simple integral is independent of the chosen representation).

Proof

technique · direct

1.1

The function $s$ is one of its own admissible simple minorants, so [L1] gives $\int s d μ \geq \int_{s i m p l e} s d μ .$

givenL1L3

1.2

If $u$ is any admissible simple minorant of $s$ , then $u \leq s$ , so [L2] gives $\int_{s i m p l e} u d μ \leq \int_{s i m p l e} s d μ .$ Taking the supremum over all such $u$ in [L1] yields the reverse inequality.

L1L2L3

2.1

The two inequalities from steps 1.1 and 1.2 are equalities, so the two [step 1.1, step 1.2] ∎ integrals agree on simple functions.

Depends on

Used by

Additivity of the nonnegative Lebesgue integral Corollary
Integrating against counting measure recovers a series Example
Monotonicity and nonnegative homogeneity of the nonnegative integral Proposition
Integrating against a density agrees with integrating the product Theorem
Monotone convergence for the integral Theorem

Dependency tree · two levels

7 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 4.4 (standard reference, not scraped)