structural:scholar_edition, stylistic:style_transfer, structural:reformatting, linguistic:translation
Measure theory

In measure theory, length is most often generalized to general sets of R n {\displaystyle \mathbb {R} ^{n}} via the Lebesgue measure. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an open interval is first defined as

so that the Lebesgue outer measure μ ∗ (E) {\displaystyle \mu ^{*}(E)} of a general set E {\displaystyle E} may then be defined as