User:Ulf Rehmann/sandbox
Measure algebra may refer to:
- algebra of measures on a topological group with the operation of convolution; see measure algebra (harmonic analysis);
- normed Boolean algebra, either in general or consisting of equivalence classes of measurable sets; see measure algebra (measure theory).
Measure algebra (measure theory)
Category:Classical measure theory
$\newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\F}{\mathcal F} \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] and [H1, Sect. 40].
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.)
On motivation
Measure algebras are "a coherent way to ignore the sets of measure $0$ in a measure space" [P, 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." [H2, page 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 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 [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.
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. All separable nonatomic normalized measure algebras are mutually isomorphic.
Here "normalized" means $\mu(\bsone_{\B})=1$.
References
[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 |
[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). Zbl 0722.60001 |
[K] | Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995). MR1321597 Zbl 0819.04002 |
[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 |
[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 |
Ulf Rehmann/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/sandbox&oldid=21730