Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue measurable; the measure of the Lebesgue measurable set A is denoted by λ(A).

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.

The Lebesgue measure is often denoted dx, but this should not be confused with the distinct notion of a volume form.

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

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