TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Fubini's theorem on term-by-term differentiation for pointwise sums of nondecreasing functions

Statement

Assume the Axiom of Countable Choice.

Let $(F_{n})_{n \geq 1}$ be a sequence (Sequences of reals: bounded, eventually, frequently, tails, subsequences) of nondecreasing functions on $[a, b]$ , and suppose that the pointwise sum

F (x) : = \sum_{n \geq 1} F_{n} (x)

converges to a finite real number for every $x \in [a, b]$ . Then $F$ is nondecreasing, and for almost every $x \in (a, b)$ ,

F^{'} (x) = \sum_{n \geq 1} F_{n}^{'} (x) .

Facts & Assumptions

Given: Countable choice and a pointwise convergent series of nondecreasing functions $F = \sum_{n \geq 1} F_{n}$ on $[a, b]$ .

[A1]

The symbols are those of the statement.

Proof

technique · direct

1.1given

Every partial sum $G_{N} : = \sum_{n = 1}^{N} F_{n}$ and every tail $H_{N} : = \sum_{n > N} F_{n} = F - G_{N}$ is nondecreasing. By A monotone function is differentiable almost everywhere by the rising-sun route, the derivatives of $F$ , all $F_{n}$ , all $G_{N}$ , and all $H_{N}$ exist on a common full-measure set. On that set, $G_{N}^{'} = \sum_{n = 1}^{N} F_{n}^{'}$ and $H_{N}^{'} = F^{'} - G_{N}^{'}$ .

2.1step 1.1

Fix $N$ . Since $H_{N}$ is increasing, the derivative bound theorem For a nondecreasing function, the derivative is measurable and integrable and its integral is bounded by the total increase gives $0 \leq \int_{a}^{b} (F^{'} - \sum_{n = 1}^{N} F_{n}^{'}) d λ = \int_{a}^{b} H_{N}^{'} (x) d λ (x) \leq H_{N} (b) - H_{N} (a)$ . But $H_{N} (b) - H_{N} (a) \to 0$ because the series defining $F (a)$ and $F (b)$ both converge.

3.1step 2.1

The nonnegative functions $F^{'} - \sum_{n = 1}^{N} F_{n}^{'}$ decrease pointwise almost everywhere to $F^{'} - \sum_{n \geq 1} F_{n}^{'}$ . Step 2.1 therefore forces $\int_{a}^{b} (F^{'} - \sum_{n \geq 1} F_{n}^{'}) d λ = 0$ . Since the integrand is nonnegative, it vanishes almost everywhere. Thus $F^{'} = \sum_{n \geq 1} F_{n}^{'}$ almost everywhere.

4.1step 1.1step 2.1step 3.1∎

Steps 1.1 through 3.1 prove the theorem.

Depends on

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Sources

A. M. Bruckner, J. B. Bruckner, and B. S. Thomson, Real Analysis, 2nd ed. (standard reference, not scraped)