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I'm trying to prove that if A is a null subset of $\mathbb{R^n}$ and B is a subset of $\mathbb{R^m}$, then A x B is a null subset of $\mathbb{R^{n + m}}$. But I'm stuck with this prof, does anyone know how to prove it?

Andrés E. Caicedo
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C_Marco
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    By null subset you mean a set of measure zero, and not the empty set, correct? – Matthew Leingang Apr 23 '20 at 17:58
  • Yes, as you said, A is a subset of measure zero and I think that A x B should be a subset of measure zero, but I can't prove it. – C_Marco Apr 23 '20 at 18:13
  • Do you mean $\Bbb{R}^{n+m}$ instead of $\Bbb{R}^{n \times m}$? Also, what is your definition of a null subset? Depending on that definition, you might be able to prove it directly, only using that definition. – PhoemueX Apr 23 '20 at 19:30
  • yes, that is a mistake, I mean $\mathbb{R^{n + m}}$, thanks for pointing it. And my definition for a null set is:

    Let A be a subset of $\mathbb{R^n}$, A is null if given $\epsilon > 0$ there is a succession of rectangles fited (I don't know if the translation of fited is correct, sorry) and open n-dimensional $ { I^{(j)} } _{j=1}^\infinity $ such that:

    1. A is in the union of the $I^{(j)}$
    2. summatory of the measure of the $I^{(j)}$ < $\epsilon$
     – C_Marco Apr 23 '20 at 20:50
  • As a quick hint, it may be easier to first consider the case when $A$ is countable (or even finite). – Noah Schweber Apr 23 '20 at 22:12
  • Thanks, I will try that approach. – C_Marco Apr 24 '20 at 06:25

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