Measure algebra (measure theory)
2010 Mathematics Subject Classification: Primary: 28A60 [MSN][ZBL]
$\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 [H2, p. 43] and [K, Sect. 17.F] to $\mu(\bsone_{\B})<\infty$ (that is, $\mu$ is a totally finite measure) in [G, Sect. 2.1] to $\mu(\bsone_{\B})\le\infty$ in [P, Sect. 1.4C], [H1, Sect. 40], [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 equality mod 0. Sets of the original σ-algebra or its completion give the same result.)
Contents
On motivation
Measure algebras are "a coherent way to ignore the sets of measure $0$ in a measure space" [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." [H2, p. 42]
Probability theory without sets of probability zero (in particular, in terms of measure algebras), proposed long ago [S], [D], is "more in agreement with the historical and conceptual development of probability theory" [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 [S, p. 727] and [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)." [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?!" [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." [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 countable (anti)chain condition; being also σ-complete, it is 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 complete metric space. If it is separable, the measure algebra $(\B,\mu)$ is also called separable.
A Boolean subalgebra of $\B$ is 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 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 [C]; see also [HN, Sect. 1] and [H1, Sect. 41] (proofs); [P, Sect. 1.4] (no proof); [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 [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 [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?" [F, Vol. 3, Chap. 34, p. 162; see also pp. 174, 182].
Significantly, for standard probability spaces it is always representable. Thus, "a cavalier attitude toward sets of measure 0 can be forgiven" [P, Sect. 1.4C, p.17].
References
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[H1] | Halmos, P.R.; Measure theory, (1950), Van Nostrand, MR0033869, Zbl 0040.16802 |
[H2] | Halmos, P.R.; Lectures on ergodic theory, (1956), Math. Soc. Japan, MR0097489, Zbl 0073.09302 |
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[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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