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Given a subset $E\subseteq\mathbb{R}$, with the length of a closed Interval $I = [a,b] $ given by $l(I)=b - a$, the Lebesgue outer measure $\lambda^*(E)$ is defined as

$\lambda^*(E) = \operatorname{inf} \{ \sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of closed intervals with } E \subset \bigcup_{k=1}^\infty I_k \}$.

One might define a similar notion (I'll call it $\beta$) by $\beta(E) = \operatorname{sup} \{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of closed intervals with} \bigcup_{k=1}^\infty I_k \subset E \}$. I would have thought this would be called "Lebesgue inner measure" but that seems to mean something different.

1) This notion $\beta$ is conspicuously missing from textbooks. Why?

My guess is that maybe $\beta(I)> l(I)$ for some closed interval $I$? I say that because the proof that that can't happen with Lebesgue outer measure (i.e. that $l^*(I)=l(I)$) uses the Heine-Borel Theorem which requires that the collection of closed intervals be a cover of $I$.

2) Are there nice sufficient conditions for $\beta(E)$ to be equal to the Lebesgue outer measure $l^*(E)$? For example, are they equal if E is Lebesgue measurable?

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Because then $\beta({\bf{Q}}^{c}\cap[0,1])=-\infty$ but the inner measure of ${\bf{Q}}^{c}\cap[0,1]$ coincides with the outer measure, which the value is $1$.

We should take $\sup\emptyset=-\infty$.

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