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Difference between revisions of "Lebesgue space"

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A [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579101.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579102.png" /> is a set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579103.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579104.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579105.png" />, called measurable sets, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579106.png" /> is a measure defined on the measurable sets), isomorphic to the  "standard model" , consisting of an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579107.png" /> and an at most countable set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579108.png" /> (in  "extreme"  cases this  "model"  may consists of just the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l0579109.png" /> or of just the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791010.png" />) endowed with the following measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791011.png" />: on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791012.png" /> one takes the usual [[Lebesgue measure|Lebesgue measure]], and to each of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791013.png" /> one ascribes a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791014.png" />; the measure is assumed to be normalized, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791015.png" />. The  "isomorphism"  can be understood here in the strict sense or modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791016.png" />; one obtains, respectively, a narrower or wider version of the concept of a Lebesgue space (in the latter case one can talk about a Lebesgue space modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791018.png" />). One can give a definition of a Lebesgue space in terms of  "intrinsic"  properties of the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791019.png" /> (see [[#References|[1]]]–[[#References|[3]]]).
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A [[Measure space|measure space]] $(M,\mathfrak B, \mu)$  (where $M$ is a set, $\mathfrak B$ is a $\sigma$-algebra of subsets of $M$, called measurable sets, and $\mu$ is a measure defined on the measurable sets), isomorphic to the  "standard model" , consisting of an interval $\Delta$ and an at most countable set of points $\alpha_i$ (in  "extreme"  cases this  "model"  may consists of just the interval $\Delta$ or of just the points $\alpha_i$) endowed with the following measure $\mathfrak m$: on $\Delta$ one takes the usual [[Lebesgue measure|Lebesgue measure]], and to each of the points $\alpha_i$ one ascribes a measure $\mathfrak(\alpha_i) = \mathfrak m_i$; the measure is assumed to be normalized, that is, $\mu(M) = \mathfrak m(\Delta) + \sum\mathfrak m_i = 1$. The  "isomorphism"  can be understood here in the strict sense or modulo $0$; one obtains, respectively, a narrower or wider version of the concept of a Lebesgue space (in the latter case one can talk about a Lebesgue space modulo $0$). One can give a definition of a Lebesgue space in terms of  "intrinsic"  properties of the measure space $(M,\mathfrak B, \mu)$ (see [[#References|[1]]]–[[#References|[3]]]).
  
A Lebesgue space is the most frequently occurring type of space with a normalized measure, since any complete separable metric space with a normalized measure (defined on its Borel subsets and then completed in the usual way) is a Lebesgue space. Apart from properties common to all measure spaces, a Lebesgue space has a number of specific  "good"  properties. For example, any automorphism of a Boolean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791020.png" />-algebra on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791021.png" /> is generated by some [[Automorphism|automorphism]] of a Lebesgue space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791022.png" />. Under a number of natural operations, from a Lebesgue space one again obtains a Lebesgue space. Thus, a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791023.png" /> of positive measure in a Lebesgue space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791024.png" /> is itself a Lebesgue space (its measurable subsets are assumed to be those that are measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791025.png" />, and the measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057910/l05791026.png" />); the direct product of finitely or countably many Lebesgue spaces is a Lebesgue space. Other properties of Lebesgue spaces are connected with measurable partitions (cf. [[Measurable decomposition|Measurable decomposition]]).
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A Lebesgue space is the most frequently occurring type of space with a normalized measure, since any complete separable metric space with a normalized measure (defined on its Borel subsets and then completed in the usual way) is a Lebesgue space. Apart from properties common to all measure spaces, a Lebesgue space has a number of specific  "good"  properties. For example, any automorphism of a Boolean $\sigma$-algebra on a measure space $(\mathfrak B, \mu)$ is generated by some [[Automorphism|automorphism]] of a Lebesgue space $M$. Under a number of natural operations, from a Lebesgue space one again obtains a Lebesgue space. Thus, a subset $A$ of positive measure in a Lebesgue space $M$ is itself a Lebesgue space (its measurable subsets are assumed to be those that are measurable in $M$, and the measure is $\mu_A(X)=\mu(X) / \mu(A)$); the direct product of finitely or countably many Lebesgue spaces is a Lebesgue space. Other properties of Lebesgue spaces are connected with measurable partitions (cf. [[Measurable decomposition|Measurable decomposition]]).
  
 
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Revision as of 13:55, 4 December 2012

A measure space $(M,\mathfrak B, \mu)$ (where $M$ is a set, $\mathfrak B$ is a $\sigma$-algebra of subsets of $M$, called measurable sets, and $\mu$ is a measure defined on the measurable sets), isomorphic to the "standard model" , consisting of an interval $\Delta$ and an at most countable set of points $\alpha_i$ (in "extreme" cases this "model" may consists of just the interval $\Delta$ or of just the points $\alpha_i$) endowed with the following measure $\mathfrak m$: on $\Delta$ one takes the usual Lebesgue measure, and to each of the points $\alpha_i$ one ascribes a measure $\mathfrak(\alpha_i) = \mathfrak m_i$; the measure is assumed to be normalized, that is, $\mu(M) = \mathfrak m(\Delta) + \sum\mathfrak m_i = 1$. The "isomorphism" can be understood here in the strict sense or modulo $0$; one obtains, respectively, a narrower or wider version of the concept of a Lebesgue space (in the latter case one can talk about a Lebesgue space modulo $0$). One can give a definition of a Lebesgue space in terms of "intrinsic" properties of the measure space $(M,\mathfrak B, \mu)$ (see [1][3]).

A Lebesgue space is the most frequently occurring type of space with a normalized measure, since any complete separable metric space with a normalized measure (defined on its Borel subsets and then completed in the usual way) is a Lebesgue space. Apart from properties common to all measure spaces, a Lebesgue space has a number of specific "good" properties. For example, any automorphism of a Boolean $\sigma$-algebra on a measure space $(\mathfrak B, \mu)$ is generated by some automorphism of a Lebesgue space $M$. Under a number of natural operations, from a Lebesgue space one again obtains a Lebesgue space. Thus, a subset $A$ of positive measure in a Lebesgue space $M$ is itself a Lebesgue space (its measurable subsets are assumed to be those that are measurable in $M$, and the measure is $\mu_A(X)=\mu(X) / \mu(A)$); the direct product of finitely or countably many Lebesgue spaces is a Lebesgue space. Other properties of Lebesgue spaces are connected with measurable partitions (cf. Measurable decomposition).

References

[1] P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics. II" Ann. of Math. , 43 : 2 (1942) pp. 332–350
[2] V.A. Rokhlin, "On mean notions of measure theory" Mat. Sb. , 25 : 1 (1949) pp. 107–150 (In Russian)
[3] J. Haezendonck, "Abstract Lebesgue–Rokhlin spaces" Bull. Soc. Math. Belg. , 25 : 3 (1973) pp. 243–258


Comments

Cf. also [a1] for a discussion of Lebesgue spaces and measurable partitions, including an intrinsic description of Lebesgue spaces.

References

[a1] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Appendix 1 (Translated from Russian)
How to Cite This Entry:
Lebesgue space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lebesgue_space&oldid=29074
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article