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Linearity of convergent improper integrals

Statement

If the improper integrals of $f$ and $g$ converge over the same one-ended interval and $r,s\in\mathbb R$ , then $\int(rf+sg)=r\int f+s\int g.$ The same formula holds for mixed improper integrals when every singular-end piece of both integrals converges separately.

Facts & Assumptions

Given: Convergent improper integrals of $f$ and $g$ on the indicated domain, and scalars $r,s$ .

[L1]

Proper Riemann integration is linear (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$ ).

[L2]

Infinite and one-sided endpoint limits use the usual epsilon definitions (Limits at $+\infty$ and $-\infty$ , and infinite limits at a point, The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ ).

[L3]

Mixed convergence is defined separately on every singular piece (Improper integrals with several singular ends).

Proof

technique · direct

1.1

On every compact truncation, [L1] gives $\int(rf+sg)=r\int f+s\int g$ . Let the two truncation integrals tend to $A$ and $B$ . Given $\varepsilon>0$ , [L2] makes their respective errors smaller than $\varepsilon/(2(1+|r|))$ and $\varepsilon/(2(1+|s|))$ sufficiently near the end. The triangle inequality then makes the error of the linear combination from $rA+sB$ smaller than $\varepsilon$ . This proves the formula on every one-ended interval, including $r=0$ or $s=0$ .

L1L2

2.1

For a mixed integral, apply step 1.1 to every separately convergent piece and then add the finitely many resulting identities as required by [L3]. No assertion is made when either side would contain an indeterminate difference of divergent quantities.

L3step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)