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A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections

Statement

Let a<ba<b, let α,β:[a,b]→R\alpha,\beta:[a,b]\to\mathbb R be continuous with α≤β\alpha\le\beta, and put K:={(x,y):a≤x≤b, α(x)≤y≤β(x)}.K:=\{(x,y):a\le x\le b,\ \alpha(x)\le y\le\beta(x)\}. Then KK is compact and Jordan measurable. If a function on the open region between the graphs extends to a continuous H:K→RH:K\to\mathbb R, then HH is Riemann integrable over KK and ∫KH=∫ab(∫α(x)β(x)H(x,y) dy)dx.\int_KH=\int_a^b\left(\int_{\alpha(x)}^{\beta(x)}H(x,y)\,dy\right)dx. The formula includes coincident graphs and uses the continuous extension on the boundary.

Facts & Assumptions

Given: Continuous α≤β\alpha\le\beta on [a,b][a,b], the closed region KK, and a continuous H:K→RH:K\to\mathbb R.

[L1]

Jordan--Fubini integrates a bounded integrable function over a Jordan set by its Jordan sections (Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable).

[L2]

A metric-bounded set is Jordan measurable if and only if its boundary is null, equivalently content zero (A bounded set in Rm\mathbb{R}^m is Jordan measurable iff its boundary is null, equivalently of content zero).

[L3]

The graph of a continuous real function on a compact Jordan domain has content zero (The graph of a continuous function on a closed nondegenerate rectangle in Rm\mathbb{R}^m has content zero in Rm+1\mathbb{R}^{m+1}).

[L4]

A continuous real function on a compact Jordan set is Riemann integrable there (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set).

[L5]

A closed bounded subset of R2\mathbb R^2 is compact (Heine-Borel in Rn\mathbb{R}^n: with the Euclidean metric a subset of Rn\mathbb{R}^n is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct
1.1

The boundary of KK is contained in the graphs of α\alpha and β\beta together with the two vertical endpoint segments. Each is a continuous graph, after exchanging coordinates for the vertical segments, and hence has content zero by [L3]. The set KK is closed and bounded, hence compact by [L5], and [L2] makes it Jordan measurable.

L2L3L5given
2.1

The continuous HH is integrable on the compact Jordan set by [L4]. Every vertical section is the closed interval [α(x),β(x)][\alpha(x),\beta(x)], and its restriction is continuous, so [L1] gives the displayed formula.

L1L4step 1.1
3.1

If α(x)=β(x)\alpha(x)=\beta(x), that section is degenerate and contributes 00; the endpoint sections and all other boundary changes have content zero. Requiring a continuous extension to KK supplies boundedness and integrability that continuity only on the open region would not supply.

step 2.1algebra∎

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