|
|
| (107 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| − | "'''Measure algebra'''" may refer to:
| + | <ref> [http://hea-www.harvard.edu/AstroStat http://hea-www.harvard.edu/AstroStat]; <nowiki> http://www.incagroup.org </nowiki>; <nowiki> http://astrostatistics.psu.edu </nowiki> </ref> |
| | | | |
| − | * algebra of measures on a topological group with the operation of convolution; see [[measure algebra (harmonic analysis)]];
| + | ====Notes==== |
| − | * normed Boolean algebra, either abstract or consisting of equivalence classes of measurable sets; see [[measure algebra (measure theory)]].
| + | <references /> |
| | | | |
| − | =Measure algebra (measure theory)=
| + | ------------------------------------------- |
| | | | |
| − | {{MSC.|28A60}}
| |
| | | | |
| − | [[:Category:Classical measure theory]]
| + | {| |
| − | | + | | A || B || C |
| − | {{TEX|done}}
| + | |- |
| − | | + | | X || Y || Z |
| − | $\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 {{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|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}}
| |
| − | | |
| − | 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}}
| |
| − | | |
| − | 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.''
| |
| − | | |
| − | 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 $\mu(\bsone_{\B})=1$.
| + | ----------------------------------------- |
| | + | ----------------------------------------- |
| | | | |
| − | 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)}}.
| + | $\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$ |
| | | | |
| − | 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"}}.
| + | <asy> |
| | + | size(100,100); |
| | + | label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0)); |
| | + | </asy> |
| | | | |
| − | ====Realization of homomorphisms====
| + | <asy> |
| | + | size(220,220); |
| | | | |
| − | 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 $B_2$.
| + | import math; |
| | | | |
| − | 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}}.
| + | int kmax=40; |
| | | | |
| − | 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}}.
| + | guide g; |
| | + | for (int k=-kmax; k<=kmax; ++k) { |
| | + | real phi = 0.2*k*pi; |
| | + | real rho = 1; |
| | + | if (k!=0) { |
| | + | rho = sin(phi)/phi; |
| | + | } |
| | + | pair z=rho*expi(phi); |
| | + | g=g..z; |
| | + | } |
| | + | |
| | + | draw (g); |
| | | | |
| − | ====References====
| + | defaultpen(0.75); |
| | + | draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) ); |
| | + | dot ( (1,0) ); |
| | + | label ( "$a$", (1,0), NE ); |
| | | | |
| − | {|
| + | </asy> |
| − | |valign="top"|{{Ref|P}}|| Karl Petersen, "Ergodic theory", Cambridge (1983). {{MR|0833286}} {{ZBL|0507.28010}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|H1}}|| P.R. Halmos, "Measure theory", Van Nostrand (1950). {{MR|0033869}} {{ZBL|0040.16802}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|H2}}|| P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956). {{MR|0097489}} {{ZBL|0073.09302}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|G}}|| Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003). {{MR|1958753}} {{ZBL|1038.37002}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|K}}|| Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995). {{MR|1321597}} {{ZBL|0819.04002}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|F}}|| D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004 {{MR|2462519}} {{ZBL|1162.28001}}; Vol. 2: 2003 {{MR|2462280}} {{ZBL|1165.28001}}; Vol. 3: 2004 {{MR|2459668}} {{ZBL|1165.28002}}; Vol. 4: 2006 {{MR|2462372}} {{ZBL|1166.28001}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|S}}|| I.E. Segal, "Abstract probability spaces and a theorem of Kolmogoroff", ''Amer. J. Math.'' '''76''' (1954), 721–732. {{MR|0063602}} {{ZBL|0056.12301}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|D}}|| L.E. Dubins, "Generalized random variables", ''Trans. Amer. Math. Soc.'' '''84''' (1957), 273–309. {{MR|0085326}} {{ZBL|0078.31003}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|W}}|| David Williams, "Probability with martingales", Cambridge (1991). {{MR|1155402}} {{ZBL|0722.60001}}
| |
| − | |-
| |
| − | |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. {{MR|1556820}} {{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. {{MR|0006617}} {{ZBL|0063.01888}}
| |
| − | |}
| |
