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A nonnegative improper Riemann integral on a half-line agrees with the Lebesgue integral

Statement

Assume the Axiom of Countable Choice. Let $a \in R$ and let $f : [a, \infty) \to [0, \infty)$ be Riemann integrable on every compact interval $[a, R]$ with $R > a$ . If the improper Riemann integral $\int_{a}^{\infty} f (x) d x$ converges in the sense of Improper integrals over unbounded intervals, then $f$ is Lebesgue integrable on $[a, \infty)$ and $\int_{[a, \infty)} f d λ_{1} = \int_{a}^{\infty} f (x) d x .$

Facts & Assumptions

Given: The Axiom of Countable Choice, a real $a$ , a nonnegative function $f : [a, \infty) \to [0, \infty)$ that is Riemann integrable on every $[a, R]$ with $R > a$ , and a finite improper Riemann integral $L : = \int_{a}^{\infty} f (x) d x$ .

[L1]

On every compact interval, a bounded Riemann integrable function is Lebesgue integrable there with the same value. (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral)

[L2]

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

[L3]

The improper integral over $[a, \infty)$ is the limit of the truncated Riemann integrals as the right endpoint tends to $+ \infty$ . (Improper integrals over unbounded intervals)

Proof

technique · direct

1.1

For each natural number $n \geq 1$ , define $f_{n} : = f χ_{[a, a + n]} .$ Then $0 \leq f_{n} \leq f_{n + 1}$ and $f_{n} (x) ↑ f (x)$ for every $x \geq a$ . By [L1] applied on $[a, a + n]$ , $\int_{[a, \infty)} f_{n} d λ_{1} = \int_{[a, a + n]} f d λ_{1} = \int_{a}^{a + n} f (x) d x .$

L1construct

2.1

Since $f_{n} ↑ f$ , [L2] gives $\int_{[a, \infty)} f d λ_{1} = \lim_{n \to \infty} \int_{[a, \infty)} f_{n} d λ_{1} = \lim_{n \to \infty} \int_{a}^{a + n} f (x) d x .$ Because $a + n \to + \infty$ , [L3] identifies the last limit with the given improper integral $L$ . Hence $\int_{[a, \infty)} f d λ_{1} = L,$ and in particular the Lebesgue integral is finite, so $f$ is integrable on the half-line.

step 1.1L2L3∎

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Sources

Richard F. Bass, Real Analysis for Graduate Students, Version 5.0, Exercise 9.4 (standard reference, not scraped)
Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis, Theorem (5.53) (standard reference, not scraped)