Editing Lebesgue measure (section)

== Properties ==
[[File:Translation of a set.svg|thumb|300px|Translation invariance: The Lebesgue measure of <math>A</math> and <math>A+t</math> are the same.]]
The Lebesgue measure on <math>\mathbb{R}^n</math> has the following properties:

# If <math display="inline">A</math> is a [[cartesian product]] of [[interval (mathematics)|intervals]] <math>I_1 \times I_2 \times ... \times I_n</math>, then ''A'' is Lebesgue-measurable and <math>\lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|.</math>
# If ''<math display="inline">A</math>'' is a union of [[countable|countably many]] pairwise disjoint Lebesgue-measurable sets, then ''<math display="inline">A</math>'' is itself Lebesgue-measurable and ''<math display="inline">\lambda(A)</math>'' is equal to the sum (or [[infinite series]]) of the measures of the involved measurable sets.
# If ''<math display="inline">A</math>'' is Lebesgue-measurable, then so is its [[Complement (set theory)|complement]].
# ''<math display="inline">\lambda(A) \geq 0</math>'' for every Lebesgue-measurable set ''<math display="inline">A</math>''.
# If ''<math display="inline">A</math>'' and''<math display="inline">B</math>'' are Lebesgue-measurable and ''<math display="inline">A</math>'' is a subset of ''<math display="inline">B</math>'', then ''<math display="inline">\lambda(A) \leq \lambda(B)</math>''. (A consequence of 2.)
# Countable [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] of Lebesgue-measurable sets are Lebesgue-measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions does not need to be closed under countable unions: <math>\{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}</math>.)
# If ''<math display="inline">A</math>'' is an [[open set|open]] or [[closed set|closed]] subset of <math>\mathbb{R}^n</math> (or even [[Borel set]], see [[metric space]]), then ''<math display="inline">A</math>'' is Lebesgue-measurable.
# If ''<math display="inline">A</math>'' is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure. 
# A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set.  This property has been used as an alternative definition of Lebesgue measurability.  More precisely, <math>E\subset \mathbb{R}</math> is Lebesgue-measurable if and only if for every <math>\varepsilon>0</math> there exist an open set <math>G</math> and a closed set <math>F</math> such that <math>F\subset E\subset G</math> and <math>\lambda(G\setminus F)<\varepsilon</math>.<ref>{{Cite book|title=Real Analysis|last=Carothers|first=N. L.|publisher=Cambridge University Press|year=2000|isbn=9780521497565|location=Cambridge|pages=[https://archive.org/details/realanalysis0000caro/page/293 293]|url=https://archive.org/details/realanalysis0000caro/page/293}}</ref>
# A Lebesgue-measurable set can be "squeezed" between a containing [[Gδ set|{{Math|G <sub>δ</sub>}} set]] and a contained [[Fσ set|{{Math|F <sub>σ</sub>}}]]. I.e, if ''<math display="inline">A</math>'' is Lebesgue-measurable then there exist a [[Gδ set|{{Math|G <sub>δ</sub>}} set]] ''<math display="inline">G</math>'' and an [[Fσ set|{{Math|F <sub>σ</sub>}}]] ''<math display="inline">F</math>'' such that ''<math display="inline">F \subseteq A \subseteq G</math>'' and ''<math display="inline">\lambda(G \setminus A) = \lambda (A \setminus F) = 0</math>''.
# Lebesgue measure is both [[Locally finite measure|locally finite]] and [[Inner regular measure|inner regular]], and so it is a [[Radon measure]].
# Lebesgue measure is [[Strictly positive measure|strictly positive]] on non-empty open sets, and so its [[Support (measure theory)|support]] is the whole of <math>\mathbb{R}^n</math>.
# If ''<math display="inline">A</math>'' is a Lebesgue-measurable set with ''<math display="inline">\lambda(A) = 0</math>'' ''(a [[null set]]), ''then every subset of ''<math display="inline">A</math>'' is also a null set. [[A fortiori|''A fortiori'']], every subset of A is measurable.
# If ''<math display="inline">A</math>'' is Lebesgue-measurable and ''x'' is an element of <math>\mathbb{R}^n</math>, then the ''translation of <math display="inline">A</math>'' ''by <math display="inline">x</math>'', defined by <math>A + x := \{a + x: a \in A\}</math>, is also Lebesgue-measurable and has the same measure as ''<math display="inline">A</math>''.
# If ''<math display="inline">A</math>'' is Lebesgue-measurable and <math>\delta>0</math>, then the ''dilation of <math>A</math> by <math>\delta</math>'' defined by <math>\delta A=\{\delta x:x\in A\}</math> is also Lebesgue-measurable and has measure <math>\delta^{n}\lambda\,(A).</math>
# More generally, if ''<math display="inline">T</math>'' is a [[linear transformation]] and ''<math display="inline">A</math>'' is a measurable subset of <math>\mathbb{R}^n</math>, then ''<math display="inline">T(A)</math>'' is also Lebesgue-measurable and has the measure <math>\left|\det(T)\right| \lambda(A)</math>.

All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following):

{{block indent|The Lebesgue-measurable sets form a [[sigma-algebra|{{mvar|σ}}-algebra]] containing all products of intervals, and <math>\lambda</math> is the unique [[Complete measure|complete]] [[translational invariance|translation-invariant]] [[measure (mathematics)|measure]] on that {{mvar|σ}}-algebra with <math>\lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.</math>}}

The Lebesgue measure also has the property of being [[Σ-finite measure|{{mvar|σ}}-finite]].