Definition:Lebesgue Measure
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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.
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
Below characterisation of Lebesgue measure $\lambda^1$ is reminiscent of the proof of Carathéodory's Theorem (Measure Theory) as discussed on User:Lord_Farin/Sandbox/Archive, and I wonder if it adds anything
Until this has been finished, please leave {{refactor}} in the code.
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.
Source of Name
This entry was named for Henri Léon Lebesgue.
