View source for Measure algebra (measure theory)

{{MSC|28A60}}

[[Category:Classical measure theory]]

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$\newcommand{\A}{\mathcal A}
\newcommand{\B}{\mathcal B} $
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}}.

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.)

====On motivation====

Measure  algebras are "a coherent way to ignore the sets of measure $0$ in a  measure space" {{Cite|P|Sect. 1.4C, p. 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|p. 42}}

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 $\B$ 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}}

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}}

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}}

====Basic notions and facts====

Let $(\B,\mu)$ be a measure algebra, and $\mu(\bsone_{\B})<\infty$.

The  Boolean algebra $\B$ satisfies the [[Chain condition#Comments|countable  (anti)chain condition]]; being also σ-complete, it is [[Boolean  algebra#complete|complete]].

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.''

{{Anchor|subalgebra}}
A Boolean [[Boolean algebra#subalgebra|subalgebra]] of $\B$ is [[Boolean algebra#regular|regular]] if and only if it is closed in the metric mentioned above; such a subalgebra (with the restriction of $\mu$) is itself a measure algebra, – a ''measure subalgebra'' of $(\B,\mu)$.

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'').

====The isomorphism theorem====

'''Theorem 1.''' All separable nonatomic normalized measure algebras are mutually [[Boolean algebra#isomorphism|isomorphic]].

Here "normalized" means $\mu(\bsone_{\B})=1$.

Therefore, all these measure algebras are isomorphic to the measure algebra of equivalence classes of Lebesgue measurable subsets of $[0,1]$.

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)}}.

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"}}.

====Realization of homomorphisms====

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  $B_2$.

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}}.

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}}.

====References====

{|
{{User:Ulf Rehmann/sandbox/Ref 
| Key = P
| Author = Petersen, Karl
| Booktitle = Ergodic theory
| Publisher = Cambridge
| Year  = 1983
| Mrn = 0833286
| Zbl = 0507.28010
| tab = true
}}
|-
{{User:Ulf Rehmann/sandbox/Ref 
| Key = H1
| Author = Halmos, P.R.
| Booktitle = Measure theory
| Publisher = Van Nostrand
| Year  = 1950
| Mrn = 0033869
| Zbl = 0040.16802
| tab = true
}}
|-
{{User:Ulf Rehmann/sandbox/Ref 
| Key = H2
| Author = Halmos, P.R.
| Booktitle = Lectures on ergodic theory
| Publisher = Math. Soc. Japan
| Year  = 1956
| Mrn = 0097489
| Zbl = 0073.09302
| tab = true
}}
|-
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|valign="top"|{{Ref|W}}|| David  Williams, "Probability with martingales", Cambridge  (1991). &nbsp;    {{MR|1155402}} &nbsp; {{ZBL|0722.60001}}
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|valign="top"|{{Ref|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.   &nbsp; {{MR|1556820}} &nbsp;  {{ZBL|0021.11403}}
|-
|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}}
|}