+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Measure and probability theory +-- {: .hide} [[!include measure theory - contents]] =-- =-- =-- # Lebesgue measure * table of contents {: toc} ## Idea The _Lebesgue measure_ is the usual [[measure]] on the [[real line]] (or on any [[Cartesian space]]). It may be characterised as the [[Radon measure]] which is translation invariant and assigns measure $1$ to the [[unit interval]] (or unit cube). It generalises (up to a scalar constant) to [[Haar measure]] on any [[locally compact space|locally compact]] [[topological group]]. ## History The Lebesgue measure's origins can be traced to the broader theory of [[Lebesgue integral|Lebesgue integration]]. The original purpose of the latter, in broad terms, was to expand the class of integrable functions in order to give meaning to functions that are _not_ [[Riemann integral|Riemann integrable]]. In order to accomplish this, the basic properties of the concept of the _length_ of an interval must be understood. This then leads to the need to fully define the concept of _measure_, particularly in relation to sets. We begin with a lemma and a corollary. +-- {: .num_lemma} ###### Lemma Let I be an interval, $I = I_{1} \cup I_{2} \cup \cdots$ where $I_{1}, I_{2}, \ldots$ are disjoint intervals. Then ${|I|} = \sum_{j=1}^{\infty} {|I_{j}|}$ (interpreted so that $\sum_{j=1}^{\infty} {|I_{j}|}$ can be $+\infty$, either because one of the summands is $+\infty$ or because the series diverges). =-- +-- {: .num_corollary} ###### Corollary If I is any interval, then $$ {|I|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : I \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\} $$ where $\{I_{j}\}$ is any countable covering of I by intervals. =-- Now suppose $B$ is an arbitrary set of real numbers. In order for $B$ to be measurable, we must have ${|B|} \leq \sum_{j=1}^{\infty} {|I_{j}|}$ where $\bigcup_{j=1}^{\infty} I_{j}$ is any countable covering of $B$ by intervals. We must also have ${|B|} \leq \bigcup_{j=1}^{\infty} {|I_{j}|}$ where the infinum is taken over all countable coverings of $B$ by intervals. ## Definition We define the __Lebesgue [[outer measure]]__ as follows: $$ {|B|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : B \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}. $$ The set $B$ is __Lebesgue measurable__ if $$ {|A|} = {|A \cap B|} + {|A \setminus B|} $$ holds for every set $A$. Restricting to these sets, Lebesgue outer measure becomes an honest [[measure]]. Note that once the Lebesgue measure is known for open sets, the outer regularity property allows us to find it for all [[Borel set]]s (but also rather more sets). ## See also * [[Jordan content]] ## References Named after [[Henri Lebesgue]]. See also * [Wikipedia](http://en.wikipedia.org/wiki/Lebesgue_measure). * [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1385) For a very intuitive and readable derivation, see: * R. Strichartz, _The Way of Analysis_, Jones & Bartlett, 2000. For a [[polynomial function]] whose Lesbegue measurability is independent of [[ZFC]], see: * [[James E. Hanson]], *Any function I can actually write down is measurable, right?* ([arXiv:2501.02693](https://arxiv.org/abs/2501.02693)) [[!redirects Lebesgue measure]] [[!redirects Lebesgue measures]] [[!redirects Lebesgue-measurable set]] [[!redirects Lebesgue-measurable sets]] [[!redirects Lebesgue measurable set]] [[!redirects Lebesgue measurable sets]] [[!redirects Lebesgue-measurable subset]] [[!redirects Lebesgue-measurable subsets]] [[!redirects Lebesgue measurable subset]] [[!redirects Lebesgue measurable subsets]]