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Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content

Statement

Let $E \subseteq R^{p + q}$ be a bounded Jordan set whose sections $E_{x}$ are Jordan measurable outside a content-zero set of parameters. Then the completed sectional-content function $x \mapsto cont (E_{x})$ is integrable and $cont (E) = \int cont (E_{x}) d x,$ with empty sections assigned content $0$ .

Consequently, if bounded Jordan sets $E, F \subseteq R^{p + q}$ have Jordan sections outside content-zero exceptional parameter sets and $cont (E_{x}) = cont (F_{x})$ wherever both are ordinary Jordan sections outside those sets, then $cont (E) = cont (F)$ .

Facts & Assumptions

Given: Bounded Jordan sets with the stated sectional hypotheses.

[L1]

Jordan--Fubini computes an integral over a bounded Jordan set by integrating its section integrals, with empty sections assigned zero (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 indicator is Riemann integrable on a bounding rectangle, and then the indicator integral is its Jordan content (A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content).

Proof

technique · direct

1.1

Apply [L1] to the constant-one function on $E$ . Its integral over $E$ is $cont (E)$ by [L2], while the integral over a section is $cont (E_{x})$ , again by [L2].

L1L2given

2.1

For $E$ and $F$ , the two completed sectional-content functions agree outside the union of their exceptional sets, which is content zero. Their integrals are therefore equal, and step 1.1 identifies those integrals with the two total contents.

L1step 1.1

3.1

Empty sections contribute $0$ . If either set has content zero, the same formula gives zero on both sides, so no nonemptiness hypothesis is hidden.

step 1.1algebra∎

Depends on

Used by

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Sources

A. Leibman, Multidimensional Real Analysis, Theorems 5.4.3-5.4.4 (standard reference, not scraped)