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Let m the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E

$m(E)=\inf \left(\sum_{j=1}^n left(\sum_{j=1}^\infty m(R_j),\:\: E\subseteq \bigcup R_j , \:\:R_j \text{ rectangles}\right)$

It is also true that lebesgue measures are regular, so $m(E)=\inf \left(m(U), E\subseteq U, \: U \text{ open set} \right)$.

Can I say that also holds $m(E)=\inf \left(\sum_{j=1}^n left(\sum_{j=1}^\infty m(B_j),\:\: E\subseteq \bigcup B_j , \:\:B_j \text{ balls}\right)$ or not?

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# Lebesgue measure of a set

Let m the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E

$m(E)=\inf \left(\sum_{j=1}^n m(R_j),\:\: E\subseteq \bigcup R_j , \:\:R_j \text{ rectangles}\right)$

It is also true that lebesgue measures are regular, so $m(E)=\inf \left(m(U), E\subseteq U, \: U \text{ open set} \right)$.

Can I say that also holds $m(E)=\inf \left(\sum_{j=1}^n m(B_j),\:\: E\subseteq \bigcup B_j , \:\:B_j \text{ balls}\right)$ or not?