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Additivity of the nonnegative Lebesgue integral

Statement

Let $(X, A, μ)$ be an arbitrary measure space. 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: An arbitrary measure space $(X, A, μ)$ ; 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

1.1L1L2construct

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

2.1step 1.1L3L4algebra∎

By [L3] and [L4], $\int (f + g) d μ = \lim_{n} \int (s_{n} + t_{n}) d μ = \lim_{n} (\int s_{n} d μ + \int t_{n} d μ) = \int f d μ + \int g d μ .$ The last limit equality also holds when either limiting integral is infinite because addition is continuous for increasing sequences in $[0, \infty]$ .

Additivity of the nonnegative Lebesgue integral

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources