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'''measure space''' is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a  [[Algebra of sets|σ-algebra]] of its subsets, and  $\mu:\A\to[0,+\infty]$ a [[measure]]. Thus, a measure space consists of a  [[measurable space]] and a measure. The notation $(X,\A,\mu)$ is often  shortened to $(X,\mu)$ and one says that $\mu$  is a measure on $X$;  sometimes the notation is shortened to $X$.
  
A [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632802.png" /> with a [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632803.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632804.png" /> (i.e. a countably-additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632805.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632806.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632807.png" />; the latter property follows from additivity if the measure is finite, i.e. does not take the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632808.png" />, or even if there is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m0632809.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328010.png" />). The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328011.png" /> is often shortened to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328012.png" /> and one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328013.png" /> is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328014.png" />; sometimes the notation is shortened to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328015.png" />. The basic case is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328016.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328017.png" />-algebra (cf. [[Algebra of sets|Algebra of sets]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328018.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328019.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328021.png" />. In this case the measure is called (totally) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328024.png" />-finite (while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328025.png" />, then it is called (totally) finite). Such is, e.g., the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328026.png" /> (cf. [[Lebesgue space|Lebesgue space]]). However, sometimes non-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328027.png" />-finite measures are encountered, such as, e.g., the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328028.png" />-dimensional [[Hausdorff measure|Hausdorff measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328030.png" />. One may also encounter modifications in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328031.png" /> takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328032.png" />, or complex or vector values, as well as cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063280/m06328033.png" /> is only finitely additive.
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====Basic notions and constructions====
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''Inner measure'' $\mu_*$ and ''outer measure'' $\mu^*$ are defined for all subsets $A\subset X$ by
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: $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
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\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$
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{{Anchor|null}}{{Anchor|full}}{{Anchor|almost}}
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$A$  is called a ''null'' (or ''negligible'') set if $\mu^*(A)=0$; in this  case the complement $X\setminus A$ is called a set of ''full measure'',  and one says that $x\notin A$ for ''almost all'' $x$ (in other words, ''almost everywhere''). Two sets $A,B\subset X$ are ''almost equal'' (or  ''equal mod 0'') if $(x\in A)\iff(x\in B)$ for almost all $x$ (in other  words, $A\setminus B$ and $B\setminus A$ are negligible). Two functions  $f,g:X\to Y$ are ''almost equal'' (or ''equal mod 0'', or ''equivalent'') if they are equal almost everywhere.
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A  subset $A\subset X$ is called ''measurable'' (or $\mu$-measurable) if it  is almost equal to some $B\in\A$. In this case  $\mu_*(A)=\mu^*(A)=\mu(B)$. If $\mu_*(A)=\mu^*(A)<\infty$ then $A$ is $\mu$-measurable. All $\mu$-measurable sets are a σ-algebra $\A_\mu$  containing $\A$.
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The ''[[Measure#product|product]]'' of two (or finitely many) measure spaces is a well-defined measure space.
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A  ''[[probability space]]'' is a measure space $(X,\A,\mu)$ satisfying  $\mu(X)=1$. The product of infinitely many probability spaces is a  well-defined probability space. (See {{Cite|D|Sect. 8.2}},  {{Cite|B|Sect. 3.5}}, {{Cite|P}}.)
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====Some classes of measure spaces====
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Let $(X,\A,\mu)$ be a measure space.
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Both  $(X,\A,\mu)$ and $\mu$ are called ''complete'' if $\A_\mu=\A$ or,  equivalently, if $\A$ contains all null sets. The ''completion'' of  $(X,\A,\mu)$ is the complete measure space $(X,\A_\mu,\tilde\mu)$ where  $\tilde\mu(A)=\mu(B)$ whenever $A\in\A_\mu$ is almost equal to $B\in\A$.
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If  $X$ is a set of finite measure, that is, $\mu(X)<\infty$, then $\mu$, and sometimes also $(X,\A,\mu)$, is called ''finite.''
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Both  $(X,\A,\mu)$ and $\mu$ are called ''σ-finite'' if $X$ can be split into  countably many sets of finite measure, that is, $X=A_1\cup  A_2\cup\dots$ for some $A_n\in\A$ such that $\forall n \;\;  \mu(A_n)<\infty$. (Finite measures are also σ-finite.)
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Let  $\mu(X)<\infty$. Both $(X,\A,\mu)$ and $\mu$ are called [[Perfect  measure|''perfect'']] if for every $\mu$-measurable (or equivalently,  for every $\A$-measurable) function $f:X\to\R$ the image $f(X)$ contains  a Borel (or equivalently, σ-compact) subset $B$ whose preimage  $f^{-1}(B)$ is of full measure. (See {{Cite|B|Sect. 7.5}}.)
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For  ''[[standard probability space]]s'' see the separate article. Standard  measure spaces are defined similarly. They are perfect, and admit a  complete classification (unlike perfect measure spaces in general).
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''Examples.''  The real line with Lebesgue measure on Borel σ-algebra is an incomplete  σ-finite measure space. The real line with Lebesgue measure on Lebesgue  σ-algebra is a complete σ-finite measure space. The unit interval  $(0,1)$ with Lebesgue measure on Lebesgue σ-algebra is a standard  probability space. The product of countably many copies of this space is  standard; for uncountably many factors the product is perfect but  nonstandard. The one-dimensional [[Hausdorff measure]] on the plane is not σ-finite.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos,   "Measure theory" , v. Nostrand (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
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{|
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|valign="top"|{{Ref|T}}||  Terence Tao, "An introduction to measure  theory", AMS (2011).  &nbsp; {{MR|2827917}} &nbsp;  {{ZBL|05952932}}
 +
|-
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|valign="top"|{{Ref|C}}|| Donald L. Cohn, "Measure theory",  Birkhäuser (1993). &nbsp;    {{MR|1454121}}  &nbsp;  {{ZBL|0860.28001}}
 +
|-
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|valign="top"|{{Ref|P}}||  David Pollard,  "A user's guide to measure theoretic probability",  Cambridge (2002).  &nbsp;  {{MR|1873379}} &nbsp;  {{ZBL|0992.60001}}
 +
|-
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|valign="top"|{{Ref|B}}|| V.I.  Bogachev, "Measure theory", Springer-Verlag (2007). &nbsp;   {{MR|2267655}}  &nbsp;{{ZBL|1120.28001}}
 +
|-
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|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability",   Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp;  {{ZBL|0686.60001}}
 +
|}

Revision as of 20:48, 19 February 2012

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A measure space is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a σ-algebra of its subsets, and $\mu:\A\to[0,+\infty]$ a measure. Thus, a measure space consists of a measurable space and a measure. The notation $(X,\A,\mu)$ is often shortened to $(X,\mu)$ and one says that $\mu$ is a measure on $X$; sometimes the notation is shortened to $X$.

Basic notions and constructions

Inner measure $\mu_*$ and outer measure $\mu^*$ are defined for all subsets $A\subset X$ by

$ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$

$A$ is called a null (or negligible) set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of full measure, and one says that $x\notin A$ for almost all $x$ (in other words, almost everywhere). Two sets $A,B\subset X$ are almost equal (or equal mod 0) if $(x\in A)\iff(x\in B)$ for almost all $x$ (in other words, $A\setminus B$ and $B\setminus A$ are negligible). Two functions $f,g:X\to Y$ are almost equal (or equal mod 0, or equivalent) if they are equal almost everywhere.

A subset $A\subset X$ is called measurable (or $\mu$-measurable) if it is almost equal to some $B\in\A$. In this case $\mu_*(A)=\mu^*(A)=\mu(B)$. If $\mu_*(A)=\mu^*(A)<\infty$ then $A$ is $\mu$-measurable. All $\mu$-measurable sets are a σ-algebra $\A_\mu$ containing $\A$.

The product of two (or finitely many) measure spaces is a well-defined measure space.

A probability space is a measure space $(X,\A,\mu)$ satisfying $\mu(X)=1$. The product of infinitely many probability spaces is a well-defined probability space. (See [D, Sect. 8.2], [B, Sect. 3.5], [P].)

Some classes of measure spaces

Let $(X,\A,\mu)$ be a measure space.

Both $(X,\A,\mu)$ and $\mu$ are called complete if $\A_\mu=\A$ or, equivalently, if $\A$ contains all null sets. The completion of $(X,\A,\mu)$ is the complete measure space $(X,\A_\mu,\tilde\mu)$ where $\tilde\mu(A)=\mu(B)$ whenever $A\in\A_\mu$ is almost equal to $B\in\A$.

If $X$ is a set of finite measure, that is, $\mu(X)<\infty$, then $\mu$, and sometimes also $(X,\A,\mu)$, is called finite.

Both $(X,\A,\mu)$ and $\mu$ are called σ-finite if $X$ can be split into countably many sets of finite measure, that is, $X=A_1\cup A_2\cup\dots$ for some $A_n\in\A$ such that $\forall n \;\; \mu(A_n)<\infty$. (Finite measures are also σ-finite.)

Let $\mu(X)<\infty$. Both $(X,\A,\mu)$ and $\mu$ are called perfect if for every $\mu$-measurable (or equivalently, for every $\A$-measurable) function $f:X\to\R$ the image $f(X)$ contains a Borel (or equivalently, σ-compact) subset $B$ whose preimage $f^{-1}(B)$ is of full measure. (See [B, Sect. 7.5].)

For standard probability spaces see the separate article. Standard measure spaces are defined similarly. They are perfect, and admit a complete classification (unlike perfect measure spaces in general).

Examples. The real line with Lebesgue measure on Borel σ-algebra is an incomplete σ-finite measure space. The real line with Lebesgue measure on Lebesgue σ-algebra is a complete σ-finite measure space. The unit interval $(0,1)$ with Lebesgue measure on Lebesgue σ-algebra is a standard probability space. The product of countably many copies of this space is standard; for uncountably many factors the product is perfect but nonstandard. The one-dimensional Hausdorff measure on the plane is not σ-finite.

References

[T] Terence Tao, "An introduction to measure theory", AMS (2011).   MR2827917   Zbl 05952932
[C] Donald L. Cohn, "Measure theory", Birkhäuser (1993).   MR1454121   Zbl 0860.28001
[P] David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).   MR1873379   Zbl 0992.60001
[B] V.I. Bogachev, "Measure theory", Springer-Verlag (2007).   MR2267655  Zbl 1120.28001
[D] Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
How to Cite This Entry:
Measure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_space&oldid=21240
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article