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My questions is:

  • Why the outer measure $m_n^*(A)$ is defined by $$ m_n^*(A)=\inf \left\{ \sum_{i=1}^{\infty} l(I_i): \mathbb{F}= \{I_1,I_2,\cdots \} \mbox{ is Lebesgue cover of } A \right\} $$ for $ A \subset \mathbb{R}^n $ ?
  • Is it true that $m_n^*(A) \leq \sum_{i=1}^{\infty}l(I_i) \leq l(I) $? For example can you compute $m_1^*(A)$, where $A=\{x \in \mathbb{R}: x \in [a,b] \}$?

I think you should know that whether $A=I$. Then at least $m_n^*(A) = l(I) $ But what is $I$? I'm aware about these many questions and apologize for that.

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$I$ is n-interval such that $I=I_1 \times I_2 \times \cdot$, where $I_i \in \mathbb{R}$. But notice that $I_i \subset \mathbb{R}^n$ in $m_n^*(A)$ is not same as $I_i \subset \mathbb{R}$( definition of $I$). –  laovultai Jan 23 '13 at 16:41
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