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Additivity of the nonnegative Lebesgue integral

Statement

If $f, g : X \to [0, + \infty]$ are measurable, then $\int (f + g) d μ = \int f d μ + \int g d μ .$

Facts & Assumptions

Given: Nonnegative measurable functions $f$ and $g$ .

[L1]

Nonnegative measurable functions admit increasing simple approximations (Every nonnegative measurable function is the increasing limit of simple measurable functions).

[L2]

The sum of two measurable nonnegative functions is measurable, and pointwise increasing limits stay measurable (Closure properties of measurable functions used by the integral).

[L3]

Monotone convergence holds for the nonnegative integral (Monotone convergence for the integral).

[L4]

On simple functions, the nonnegative integral agrees with the simple integral, and the latter is additive (The nonnegative integral agrees with the simple integral on simple functions, The simple integral is monotone, homogeneous, and additive).

Proof

technique · direct

1.1

Choose simple functions $s_{n} ↑ f$ and $t_{n} ↑ g$ by [L1]. Then [L1, L2, construct] $s_{n} + t_{n}$ is simple for each $n$ , and $s_{n} + t_{n} ↑ f + g$ by [L2].

2.1

By [L3] and [L4],[step 1.1, L3, L4, algebra] ∎ $\int (f + g) d μ = \lim_{n} \int (s_{n} + t_{n}) d μ = \lim_{n} (\int s_{n} d μ + \int t_{n} d μ) = \int f d μ + \int g d μ .$

Depends on

Used by

Almost-everywhere monotone convergence Corollary
Beppo Levi's theorem for nonnegative series Corollary
Reverse Fatou's lemma under an integrable majorant Corollary
The exponential tail function is integrable by monotone truncation and geometric comparison Example
The function x^-1/2 on (0,1] is unbounded and integrable Example
Chebyshev-Markov inequality for the integral Theorem
Dominated convergence Theorem
Integrable simple functions are dense in L¹(μ) Theorem
Integrating against a density agrees with integrating the product Theorem
The indefinite integral of a nonnegative measurable function is a measure Theorem
The Lebesgue integral is linear on L¹(μ) Theorem

Dependency tree · two levels

14 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 4.7 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.15 (standard reference, not scraped)