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Beppo Levi's theorem for nonnegative series

Statement

Let $(f_{k})$ be nonnegative measurable functions and let $S_{n} : = \sum_{k < n} f_{k}, S : = \sum_{k = 0}^{\infty} f_{k} .$ Then $\int S d μ = \sum_{k = 0}^{\infty} \int f_{k} d μ .$

Facts & Assumptions

Given: A sequence $(f_{k})$ of nonnegative measurable functions.

[L1]

Measurable nonnegative functions are closed under finite sums and increasing pointwise suprema (Closure properties of measurable functions used by the integral).

[L2]

The nonnegative integral is additive (Additivity of the nonnegative Lebesgue integral).

[L3]

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

Proof

technique · direct

1.1

Each partial sum $S_{n}$ is measurable by [L1], the sequence $(S_{n})$ is [L1, given] increasing, and $S_{n} ↑ S$ pointwise.

2.1

By [L2], $\int S_{n} d μ = \sum_{k < n} \int f_{k} d μ$ for every $n$ . Applying [step 1.1, L2, L3, algebra] ∎ [L3] to step 1.1 gives $\int S d μ = \lim_{n} \int S_{n} d μ = \lim_{n} \sum_{k < n} \int f_{k} d μ,$ which is exactly the displayed series identity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Richard F. Bass, Real Analysis for Graduate Students, Proposition 7.6 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.15 (standard reference, not scraped)