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I'm very new to undergraduate Lebesgue measure and am still having problems with it.

I'm trying to prove the following:

Let $A\subset\mathbb{R}^m$ and $B\subset\mathbb{R}^n$ and $A\times B\subset\mathbb{R}^{m+n}$.

If $A$ and $B$ are both measurable, then $A \times B$ is measurable.

If $A$ and $B$ are both measurable, $|A \times B|=|A|\cdot|B|$.

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    A related question: http://math.stackexchange.com/q/138762011-04-20

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Okay, given your definition:

If $G$ is open in $\mathbb{R}^n$ and $H$ is open in $\mathbb{R}^m$, is $G\times H$ open in $R^{m+n}$?

If $A\subseteq G\subseteq\mathbb{R}^n$ and $B\subseteq H\subseteq \mathbb{R}^m$, is $A\times B\subseteq G\times H$ true?

If $A\subseteq G$ and $B\subseteq H$, what is $(G\times H)\setminus(A\times B)$? (Careful here...) Can you make this arbitrarily small, if you can make $G-A$ and $H-B$ arbitrarily small?

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    come now, don't be so harsh. @Josh: while this may or may not be a _homework_ problem, it will benefit you to read and follow the advice [here](http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question), since the question you asked is similar in scope to one that is typically assigned for exercise. Pay especially attention to the section on what information you should include when asking a question.2011-04-21
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This is an elaboration of Arturo's answer; it is too long to be a comment. It usus your definition of measurable set and Fubini's theorem. Proceed in steps:

  1. We may assume tha $A$ and $B$ are of finite measure, say $m(A),m(B)\le M$.
  2. Let $\epsilon>0$ be given. Choose open sets $G\subset\mathbb{R}^m$, $H\subset\mathbb{R}^n$ such that $A\subset G$, $B\subset H$ and $m(G\setminus A)<\epsilon$, $m(H\setminus B)<\epsilon$; then $m(G),m(H)\le M+\epsilon$.
  3. Choose open sets $G^*\subset\mathbb{R}^m$, $H^*\subset\mathbb{R}^n$ such that $G\setminus A\subset G^*$, $H\setminus B\subset H^*$, $m(G^*\setminus(G\setminus A))<\epsilon$, $m(H^*\setminus(H\setminus B))<\epsilon$. Then $m(G^*),m(H^*)<2\epsilon$.
  4. $G\times H\setminus A\times B=A\times(H\setminus B)\cup(G\setminus A)\times H\subset G\times H^*\cup G^*\times H$.
  5. $G\times H^*$ and $G^*\times H$ are open, and hence measurable.
  6. Use some form of Fubini's theorem to show that $m(G\times H^*)=m(G)\cdot m(H^*)$ and $m(G^*\times H)=m(G^*)\cdot m(H)$.
  7. Deduce that $m^*(G\times H\setminus A\times B)\le4(M+\epsilon)\epsilon$ ($m^*$ is the exterior measure; we still do not know that $G\times H\setminus A\times B$ is measurable.)
  8. Once you know $A$ and $B$ are measurable, again Fubini's theorem will show that $m(A\times B)=m(A)\cdot m(B)$.