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Arzela's bounded convergence theorem for Riemann integrals

Statement

Assume the Axiom of Countable Choice. Let $a < b$ , let $f_{n} : [a, b] \to R$ be Riemann integrable for every $n$ , and let $f : [a, b] \to R$ be Riemann integrable. Suppose that $f_{n} (x) \to f (x)$ for every $x \in [a, b]$ and that there is a real $M \geq 0$ with $∣ f_{n} (x) ∣ \leq M$ for all $n$ and all $x \in [a, b]$ . Then $\int_{a}^{b} f_{n} (x) d x ⟶ \int_{a}^{b} f (x) d x .$

Facts & Assumptions

Given: The Axiom of Countable Choice, reals $a < b$ , Riemann integrable functions $f, f_{n} : [a, b] \to R$ with $f_{n} (x) \to f (x)$ for every $x \in [a, b]$ , and a real $M \geq 0$ with $∣ f_{n} (x) ∣ \leq M$ for all $n$ and $x$ .

[L1]

A bounded Riemann integrable function on $[a, b]$ is Lebesgue measurable, Lebesgue integrable, and has the same Lebesgue and Riemann integrals. (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral)

[L2]

On a finite measure space, almost-everywhere pointwise convergence of a uniformly bounded measurable sequence implies convergence of the integrals. (Bounded convergence on a finite measure space)

[L3]

The interval $[a, b]$ has finite Lebesgue measure $b - a$ . (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included)

Proof

technique · direct

1.1L1L3

By [L1], each $f_{n}$ and $f$ is Lebesgue measurable and integrable on $[a, b]$ , and $\int_{[a, b]} f_{n} d λ_{1} = \int_{a}^{b} f_{n} (x) d x, \int_{[a, b]} f d λ_{1} = \int_{a}^{b} f (x) d x .$ Also [L3] makes $([a, b], L ([a, b]), λ_{1})$ a finite measure space.

2.1step 1.1L2∎

The convergence hypothesis is pointwise, hence almost everywhere, and the uniform bound $∣ f_{n} ∣ \leq M$ holds everywhere. So [L2] applies on $[a, b]$ and gives $\int_{[a, b]} f_{n} d λ_{1} ⟶ \int_{[a, b]} f d λ_{1} .$ Translating the two sides with step 1.1 yields the stated convergence of the Riemann integrals.

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Sources

Richard F. Bass, Real Analysis for Graduate Students, Version 5.0, Section 9.1 (standard reference, not scraped)
Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis, Corollary (10.32) (standard reference, not scraped)