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| In [[mathematics]], a [[subset]] <math>A</math> of a [[Polish space]] <math>X</math> is '''universally measurable''' if it is [[measurable set|measurable]] with respect to every [[complete measure|complete]] [[probability measure]] on <math>X</math> that measures all [[Borel set|Borel]] subsets of <math>X</math>. In particular, a universally measurable set of [[real number|real]]s is necessarily [[Lebesgue measurable]] (see [[#Finiteness condition]]) below.
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| Every [[analytic set]] is universally measurable. It follows from [[projective determinacy]], which in turn follows from sufficient [[large cardinal]]s, that every [[projective set]] is universally measurable.
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| ==Finiteness condition==
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| The condition that the measure be a [[probability measure]]; that is, that the measure of <math>X</math> itself be 1, is less restrictive than it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, ''N''<sub>0</sub>=<nowiki>[0,1)</nowiki>, ''N''<sub>1</sub>=<nowiki>[1,2)</nowiki>, ''N''<sub>2</sub>=<nowiki>[-1,0)</nowiki>, ''N''<sub>3</sub>=<nowiki>[2,3)</nowiki>, ''N''<sub>4</sub>=<nowiki>[-2,-1)</nowiki>, and so on. Now letting μ be Lebesgue measure, define a new measure ν by
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| : <math>\nu(A)=\sum_{i=0}^\infty \frac{1}{2^{n+1}}\mu(A\cap N_i)</math>
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| Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable. More generally a universally measurable set must be measurable with respect to every [[sigma-finite]] measure that measures all Borel sets.
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| == Example contrasting with Lebesgue measurability ==
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| Suppose <math>A</math> is a subset of [[Cantor space]] <math>2^\omega</math>; that is, <math>A</math> is a set of infinite [[sequence]]s of zeroes and ones. By putting a binary point before such a sequence, the sequence can be viewed as a [[real number]] between 0 and 1 (inclusive), with some unimportant ambiguity. Thus we can think of <math>A</math> as a subset of the interval <nowiki>[0,1]</nowiki>, and evaluate its [[Lebesgue measure]]. That value is sometimes called the '''coin-flipping measure''' of <math>A</math>, because it is the [[probability]] of producing a sequence of heads and tails that is an element of <math>A</math>, upon flipping a fair coin infinitely many times.
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| Now it follows from the [[axiom of choice]] that there are some such <math>A</math> without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an <math>A</math>, the probability that the sequence of flips of a fair coin will wind up in <math>A</math> is not well-defined. This is a pathological property of <math>A</math> that says that <math>A</math> is "very complicated" or "ill-behaved".
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| From such a set <math>A</math>, form a new set <math>A'</math> by performing the following operation on each sequence in <math>A</math>: Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Now <math>A'</math> is intuitively no "simpler" or "better-behaved" than <math>A</math>. However, the probability that the sequence of flips of a fair coin will wind up in <math>A'</math> ''is'' well-defined, for the rather silly reason that the probability is zero (to get into <math>A'</math>, the coin must come up tails on every even-numbered flip).
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| For such a set of sequences to be ''universally'' measurable, on the other hand, an arbitrarily ''biased'' coin may be used--even one that can "remember" the sequence of flips that has gone before--and the probability that the sequence of its flips ends up in the set, must be well-defined. Thus the <math>A'</math> described above is ''not'' universally measurable, because we can test it against a coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.
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| == References ==
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| * {{citation
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| | author = Alexander Kechris
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| | title = Classical Descriptive Set Theory
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| | publisher = Springer
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| | series = Graduate Texts in Mathematics
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| | volume = 156
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| | year = 1995
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| | isbn = 0-387-94374-9
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| }}
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| * {{citation
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| | author = Nishiura Togo
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| | title = Absolute Measurable Spaces
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| | publisher= Cambridge University Press
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| | year= 2008
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| | isbn = 0-521-87556-0
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| }}
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| [[Category:Descriptive set theory]]
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| [[Category:Determinacy]]
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| [[Category:Measure theory]]
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Hello from Austria. I'm glad to be here. My first name is Dave.
I live in a small city called Radisch in western Austria.
I was also born in Radisch 22 years ago. Married in November 2010. I'm working at the college.
Here is my weblog - How To Get Free Fifa 15 Coins