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Let $A$ and $B$ be rectangles in $\mathbb{R}^k$ and $\mathbb{R}^n$, respectively. Let $S$ be a set contained in $A\times B$. For each $y\in B$, let $$S_y=\{x\mid x\in A, (x,y)\in S\}.$$ Show that if $S$ is Jordan-measurable, and if $S_y$ is Jordan-measurable for each $y\in B$, then $$v(S)=\int_{y\in B}v(S_y).$$

So, writing out the definition of Jordan measure, we want to prove $$\int_S1=\int_{y\in B}\int_{S_y}1.$$ (That also includes proving that $\int_{y\in B}\int_{S_y}1$ exists.)

The double integral reminds me a little bit of Fubini's theorem, but we need to apply it on integrations over rectangles. Other than that, I can't see how to go.

PJ Miller
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1 Answers1

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You are correct, it is an application of Fubini.

A bounded set $S$ is Jordan measurable iff $1_S$ is Riemann integrable.

Hence $v(S) = \int_{A \times B} 1_S((a,b)) \,da\, db $. Fubini's theorem tells us that the function $b \mapsto \int_A 1_S((a,b)) \,da$ is Riemann integrable and furthermore, $\int_{A \times B} 1_S((a,b)) \,da\, db = \int_B (\int_A 1_S((a,b)) \,da) \, db$.

We note that $v(S_y) = \int_A 1_S((a,b)) \,da$, from which we get $v(S) = \int_B v(S_y) d b$.

copper.hat
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