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Change of variable in an improper integral

Statement

Let $I$ and $J$ be intervals, possibly open at finite or infinite singular ends, and let $ϕ : I \to J$ be a monotone differentiable surjection. Assume $ϕ^{'}$ is locally Riemann integrable and the proper change-of-variable hypotheses hold on every compact truncation. If $f$ is locally Riemann integrable on $J$ , then $\int_{J} f (x) d x and \int_{I} f (ϕ (t)) ∣ ϕ^{'} (t) ∣ d t$ converge simultaneously and, when convergent, are equal. At several singular ends this assertion is applied separately to the corresponding ends; orientation is retained for decreasing parametrizations.

Facts & Assumptions

Given: The intervals, monotone surjection $ϕ$ , and locally integrable $f$ in the statement.

[L1]

The proper change-of-variable theorem gives equality on corresponding compact truncations (Monotone change of variable for Riemann-integrable functions).

[L2]

Monotonicity and surjectivity send truncations tending to an endpoint of $I$ to truncations tending to the corresponding endpoint of $J$ .

[L3]

Improper convergence at one end is the existence of a finite limit of the corresponding compact-truncation values (Improper integrals over unbounded intervals, Improper integrals at a finite singular endpoint).

[L4]

Mixed improper integrals require separate convergence at every singular end (Improper integrals with several singular ends).

Proof

technique · direct

1.1L1

On each compact source truncation $[u, v] \subset I$ , [L1] gives [L1] $\int_{ϕ ([u, v])} f = \int_{u}^{v} (f \circ ϕ) ∣ ϕ^{'} ∣,$ with the endpoint order adjusted when $ϕ$ decreases.

2.1step 1.1L2L3

By [L2], the two sides form the same family of values as the corresponding truncation approaches a singular end. Since they are equal term by term by step 1.1, the epsilon condition in [L3] holds for one family exactly when it holds for the other, with the same limit.

3.1L4step 2.1∎

For multiple ends, apply step 2.1 separately at each matched end and add only after all pieces converge, as [L4] requires. The oriented convention supplies the sign for a decreasing parametrization.

Depends on

Dependency tree · 2 levels

Sources

William F. Trench, Introduction to Real Analysis, Theorem 3.4.13