Show that each of the following two statements is equivalent...

Show that each of the following two statements is equivalent to saying that $E \in \mathcal{M}$ : (i) given $\varepsilon>0$ there is an open set $O \supset E$ with $m^{*}(O \backslash E)<\varepsilon$, (ii) given $\varepsilon>0$ there is a closed set $F \subset E$ with $m^{*}(E \backslash F)<\varepsilon$

Show that each of the following two statements is equivalent to saying that $E \in \mathcal{M}$ :
(i) given $\varepsilon>0$ there is an open set $O \supset E$ with $m^{*}(O \backslash E)<\varepsilon$,
(ii) given $\varepsilon>0$ there is a closed set $F \subset E$ with $m^{*}(E \backslash F)<\varepsilon$

Key Concepts

Lebesgue Measurability

This concept refers to the property of a set being measurable with respect to the Lebesgue measure. A set is considered Lebesgue measurable if it satisfies the Carathéodory criterion, meaning that for any other set, the measure of that other set is exactly divided by its intersection with the set in question and its complement. In the context of the problem, showing that a set can be approximated by both open supersets and closed subsets in terms of outer measure is equivalent to proving its measurability.

Outer Measure

The outer measure is a function that assigns a non-negative number to any set, often defined as the infimum of the measures of countable coverings by open intervals or sets. It is fundamental in defining Lebesgue measure since it provides a way to 'size' all subsets of the real numbers, not just the measurable ones. In the discussed equivalences, the outer measure is used to quantify the error between the approximating open or closed set and the original set.

Regularity of Lebesgue Measure

Regularity describes the ability to approximate measurable sets arbitrarily well both from the outside by open sets and from the inside by closed sets. The statements given in the question are examples of such regularity properties. They claim that every set that is Lebesgue measurable can be closely approximated in measure by a more 'nice' set (either open or closed), which is a key characteristic of the Lebesgue measure.

Open and Closed Set Approximations

This concept involves approximating any set by open supersets or closed subsets such that the difference in outer measure can be made arbitrarily small. It forms the basis of proving the equivalence between these approximations and the measurable nature of a set. In the problem, it is shown that if a set can be almost contained in an open set from above and almost contain a closed set from below, then it must be Lebesgue measurable.

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