= Regular outer measure
{title2=$\forall A\ \exists G\supseteq A:\quad G\text{ measurable},\quad\mu^*(G)=\mu^*(A)$}
An <outer measure> is regular in this sense when every <set> admits a measurable hull of equal <outer measure>. This is different from merely specifying inner or outer approximation of already <measurable sets> by compact or open sets. <Lebesgue outer measure> has a hull that is a <Borel set>: for finite <outer measure> choose open covers of total volumes within $1/j$ of its value and intersect their unions. <Monotonicity> gives equality of the hull's <outer measure> with that of the original <set>. For infinite <outer measure> the whole Euclidean space is a hull. This regularity is useful for turning numerical equality of inner and outer sizes into <measurability>.
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