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"'''Measure algebra'''" may refer to:
 
  
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* algebra of measures on a topological group with the operation of convolution; see [[measure algebra (harmonic analysis)]];
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====References====
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* normed Boolean algebra, either abstract or consisting of equivalence classes of measurable sets; see [[measure algebra (measure theory)]].
 
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=Measure algebra (measure theory)=
 
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{{MSC.|28A60}}
 
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[[:Category:Classical measure theory]]
 
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{{TEX|done}}
 
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$\newcommand{\Om}{\Omega}
 
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\newcommand{\om}{\omega}
 
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\newcommand{\F}{\mathcal F}
 
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\newcommand{\B}{\mathcal B} $
 
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A '''measure algebra''' is a pair $(\B,\mu)$ where $\B$ is a Boolean σ-algebra and $\mu$ is a (strictly) positive measure on $\B$. The (strict) positivity means $\mu(x)\ge0$ and $\mu(x)=0\iff x=\bszero_{\B}$ for all $x\in\B$. However, about the greatest value $\mu(\bsone_{\B})$ of $\mu$, assumptions differ: from $\mu(\bsone_{\B})=1$ (that is, $\mu$ is a probability measure) in {{Cite|H2|p. 43}} and {{Cite|K|Sect. 17.F}} to $\mu(\bsone_{\B})<\infty$ (that is, $\mu$ is a totally finite measure) in {{Cite|G|Sect. 2.1}} to $\mu(\bsone_{\B})\le\infty$ in {{Cite|P|Sect. 1.4C}}, {{Cite|H1|Sect. 40}}, {{Cite|F|Vol. 3, Sect. 321}}.
 
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A ''measure algebra of a [[measure space]]'' consists, by definition, of all equivalence classes of measurable sets. (The equivalence is [[Measure space#Basic notions and constructions|equality mod 0]]. Sets of the original σ-algebra or its [[Measure space#Completion|completion]] give the same result.)
 
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====On motivation====
 
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Measure algebras are "a coherent way to ignore the sets of measure $0$ in a measure space" {{Cite|P|Sect. 1.4C, page 15}}. "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure zero. The algebraic treatment gets rid of this source of unpleasantness by refusing to consider sets at all; it considers sets modulo sets of measure zero instead." {{Cite|H2|page 42}}
 
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Probability theory without sets of probability zero (in particular, in terms of measure algebras), proposed long ago {{Cite|S}}, {{Cite|D}}, is "more in agreement with the historical and conceptual development of probability theory" {{Cite|S|Introduction}}. An event is defined here as an element of a $\B$ where $(\B,\mu)$ is a measure algebra; accordingly, a random variable with values in a measurable space $(X,\A)$ is defined as a σ-homomorphism from $\A$ (treated as a Boolean σ-algebra) to $\B$; see {{Cite|S|p. 727}} and {{Cite|D|p. 273}}. "The basic conceptual concern in statistics is not so much with the values of the measurable function $f$ representing a random variable ... as with the sets ... where $f$ takes on certain values (and with the probabilities of those sets)." {{Cite|S|p. 727}}
 
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In spite of elegance and other advantages, the measure algebra approach to probability is not the mainstream. When dealing with random processes, "the equivalence-class formulation just will not work: the 'cleverness' of introducing quotient spaces loses the subtlety which is essential even for formulating the fundamental results on existence of continuous modifications, etc., unless one performs contortions which are hardly elegant. Even if these contortions allow one to ''formulate'' results, one would still have to use genuine functions to ''prove'' them; so where does the reality lie?!" {{Cite|W|p. xiii}}
 
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Bad news: we cannot get rid of measure spaces and sets of measure zero. Good news: we can get rid of pathological measure spaces, thus achieving harmony between measure spaces and measure algebras. "Since it can be argued that sets of measure zero are worthless, not only from the algebraic but also from the physical point of view, and since every measure algebra can be represented as the algebra associated with a non-pathological measure space, the poverty of some measure spaces may be safely ignored." {{Cite|H2|p. 43}}
 
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====Basic notions and facts====
 
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Let $(\B,\mu)$ be a measure algebra, and $\mu(\bsone_{\B})<\infty$.
 
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The Boolean algebra $\B$ satisfies the [[Chain condition#Comments|countable (anti)chain condition]]; being also σ-complete, it is [[Boolean algebra#complete|complete]].
 
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Defining the distance between $A,B\in\B$ as $\mu(A\Delta B)$ (the measure of their symmetric difference) one turns $B$ into a [[metric space]]. This is always a [[Metric space#complete|complete]] metric space. If it is [[Separable space|separable]], the measure algebra $(\B,\mu)$ is also called ''separable.''
 
  
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An ''atom'' of $\B$ is, by definition, an element $A\in\B$ such that $A>\bszero_{\B}$ and no $B\in\B$ satisfies $A>B>\bszero_{\B}$. If $\B$ contains no atoms it is called ''nonatomic'' (or ''atomless'').
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|valign="top"|{{Ref|I}}||  Kiyosi Itô, "Introduction to  probability theory", Cambridge (1984).  &nbsp; {{MR|0777504}}  &nbsp; {{ZBL|0545.60001}}
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|valign="top"|{{Ref|Ru}}||  Thierry de la Rue, "Espaces de Lebesgue", ''Séminaire de Probabilités  XXVII,'' Lecture Notes in Mathematics, '''1557''' (1993), Springer,  Berlin,  pp. 15–21. &nbsp;  {{MR|1308547}} &nbsp;  {{ZBL|0788.60001}}
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|valign="top"|{{Ref|H}}||  Jean  Haezendonck, "Abstract  Lebesgue-Rohlin  spaces", ''Bull. Soc.  Math.  de Belgique'' '''25'''  (1973), 243–258.  &nbsp;    {{MR|0335733}}  &nbsp;  {{ZBL|0308.60006}}
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|valign="top"|{{Ref|HN}}||  P.R. Halmos, J. von Neumann, "Operator  methods in classical mechanics,  II", ''Annals of Mathematics (2)''  '''43''' (1942), 332–350.    &nbsp;  {{MR|0006617}} &nbsp;    {{ZBL|0063.01888}}
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|valign="top"|{{Ref|Ro}}||  V.A. Rokhlin, (1962), "On the fundamental ideas of measure theory",  ''Translations (American Mathematical Society) Series 1,'' 10 (1962),  1–54. &nbsp; {{MR|0047744}} &nbsp; Translated from Russian:  Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический  Сборник (Новая Серия) 25(67): 107–150. &nbsp; {{MR|0030584}}
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====The isomorphism theorem====
 
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'''Theorem 1.''' All separable nonatomic normalized measure algebras are mutually [[Boolean algebra#isomorphism|isomorphic]].
 
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Here "normalized" means $\mu(\bsone_{\B})=1$.
 
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Theorem 1 is due to Carathėodory 1939 {{Cite|C}}; see also {{Cite|HN|Sect. 1}} and {{Cite|H1|Sect. 41}} (proofs); {{Cite|P|Sect. 1.4}} (no proof); {{Cite|K|Exercise (17.44)}}.
 
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Isomorphic classification of all totally finite (and σ-finite, and some more; not necessarily separable or nonatomic) measure algebras is available, see {{Cite|F|Vol. 3 "Measure algebras", Chapter 33 "Maharam's theorem"}}.
 
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====Realization of homomorphisms====
 
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Every measure preserving map $\phi:X_1\to X_2$ between measure spaces $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$ induces a homomorphism $\Phi:\B_2\to\B_1$ between their measure algebras $(\B_1,\nu_1)$, $(\B_2,\nu_2)$ as follows: $\Phi(B_2)$ (for $B_2\in\B_2$) is the equivalence class of the inverse image $\phi^{-1}(A_2)$ of some (therefore every) set $A_2\in\A_2$ belonging to the equivalence class $B_2$.
 
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In general, a homomorphism $\Phi:\B_2\to\B_1$ is not necessarily induced by some measure preserving map $\phi:X_1\to X_2$ (even if $(X_1,\A_1,\mu_1)=(X_2,\A_2,\mu_2)$ is a probability space and $\Phi$ is an automorphism). According to {{Cite|F}}, it is "one of the central problems of measure theory: under what circumstances will a homomorphism between measure algebras be representable by a function between measure spaces?" {{Cite|F|Vol. 3, Chap. 34, p. 162; see also pp. 174, 182}}.
 
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Significantly, for [[standard probability space]]s it is always representable. Thus, "a cavalier attitude toward sets of measure 0 can be forgiven" {{Cite|P|Sect. 1.4C, p.17}}.
 
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====References====
 
  
 
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Revision as of 20:10, 5 April 2012

References

[I] Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001
[Ru] Thierry de la Rue, "Espaces de Lebesgue", Séminaire de Probabilités XXVII, Lecture Notes in Mathematics, 1557 (1993), Springer, Berlin, pp. 15–21.   MR1308547   Zbl 0788.60001
[H] Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", Bull. Soc. Math. de Belgique 25 (1973), 243–258.   MR0335733   Zbl 0308.60006
[HN] P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics, II", Annals of Mathematics (2) 43 (1942), 332–350.   MR0006617   Zbl 0063.01888
[Ro] V.A. Rokhlin, (1962), "On the fundamental ideas of measure theory", Translations (American Mathematical Society) Series 1, 10 (1962), 1–54.   MR0047744   Translated from Russian: Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический Сборник (Новая Серия) 25(67): 107–150.   MR0030584


[P] Karl Petersen, "Ergodic theory", Cambridge (1983).   MR0833286   Zbl 0507.28010
[H1] P.R. Halmos, "Measure theory", Van Nostrand (1950).   MR0033869   Zbl 0040.16802
[H2] P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956).   MR0097489   Zbl 0073.09302
[G] Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003).   MR1958753   Zbl 1038.37002
[K] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[F] D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004   MR2462519   Zbl 1162.28001; Vol. 2: 2003   MR2462280   Zbl 1165.28001; Vol. 3: 2004   MR2459668   Zbl 1165.28002; Vol. 4: 2006   MR2462372   Zbl 1166.28001
[S] I.E. Segal, "Abstract probability spaces and a theorem of Kolmogoroff", Amer. J. Math. 76 (1954), 721–732.   MR0063602   Zbl 0056.12301
[D] L.E. Dubins, "Generalized random variables", Trans. Amer. Math. Soc. 84 (1957), 273–309.   MR0085326   Zbl 0078.31003
[W] David Williams, "Probability with martingales", Cambridge (1991).   MR1155402   Zbl 0722.60001
[C] Constantin Carathėodory, "Die homomorphieen von Somen und die Multiplikation von Inhaltsfunktionen" (German), Annali della R. Scuola Normale Superiore di Pisa (Ser. 2) 8 (1939), 105–130.   MR1556820   Zbl 0021.11403
[HN] P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics, II", Annals of Mathematics (2) 43 (1942), 332–350.   MR0006617   Zbl 0063.01888
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