Definition:Lebesgue Measure

Definition
Let $\mathcal{J}_{ho}^n$ be the set of half-open $n$-rectangles.

Let $\mathcal B \left({\R^n}\right)$ be the Borel $\sigma$-algebra on $\R^n$.

Let $\lambda^n$ be $n$-dimensional Lebesgue pre-measure on $\mathcal{J}_{ho}^n$.

Any measure $\mu$ extending $\lambda^n$ to $\mathcal B \left({\R^n}\right)$ is called $n$-dimensional Lebesgue measure.

That is, $\mu$ is an $n$-dimensional Lebesgue measure iff it satisfies:


 * $\mu \restriction_{\mathcal{J}_{ho}^n} = \lambda^n$

where $\restriction$ denotes restriction.

By virtue of Existence and Uniqueness of Lebesgue Measure, one may speak simply about (the) $n$-dimensional Lebesgue measure.

Usually, this measure is also denoted by $\lambda^n$, even though this may be considered abuse of notation.

Lebesgue Measure on the Reals
For a given set $S \subseteq \R$, let $\left\{{I_n}\right\}$ be a countable set of open intervals such that


 * $S \subseteq \bigcup I_n$

For the set of all subsets $\mathcal P \left({\R}\right)$ of the reals $\R$, construct a function $\mu^*:\mathcal P \left({\R}\right) \to \R_+$ as:


 * $\displaystyle m^*(S) = \inf \left\{{\sum_{n \mathop \in \N} l \left({I_n}\right) : \left\{{I_n}\right\} : S \subseteq \bigcup_{n \mathop \in \N} I_n}\right\}$

where the infimum ranges over all such sets $\left\{{I_n}\right\}$, and $l(I_n)$ is the length of the interval.

Then $\mu^*$ is known as the Lebesgue outer measure and can be shown to be an outer measure.

When the domain of $\mu^*$ is restricted to the set $\mathfrak M$ of Lebesgue-measurable sets, $\mu^*$ is instead written as $\mu$ and is known as the Lebesgue measure.

Moreover, $(\R, \mathfrak M, \mu)$ is a measure space.