CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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

1.1L1given

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

2.1L2step 1.1L3algebra

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.

Beppo Levi's theorem for nonnegative series

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources