Universally measurable set
Encyclopedia
In mathematics
, a subset
of a Polish space
is universally measurable if it is measurable with respect to every complete
probability measure
on
that measures all Borel
subsets of
. In particular, a universally measurable set of real
s is necessarily Lebesgue measurable (see #Finiteness condition) below.
Every analytic set
is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.
; that is, that the measure of
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, N0=[0,1) , N1=[1,2) , N2=[-1,0) , N3=[2,3) , N4=[-2,-1) , and so on. Now letting μ be Lebesgue measure, define a new measure ν by
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.
is a subset of Cantor space
; that is, 
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
as a subset of the interval [0,1] , and evaluate its Lebesgue measure
. That value is sometimes called the coin-flipping measure of
, because it is the probability
of producing a sequence of heads and tails that is an element of
, upon flipping a fair coin infinitely many times.
Now it follows from the axiom of choice that there are some such
without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an 
, the probability that the sequence of flips of a fair coin will wind up in 
is not well-defined. This is a pathological property of 
that says that 
is "very complicated" or "ill-behaved".
From such a set
, form a new set 
by performing the following operation on each sequence in 
: Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Now 
is intuitively no "simpler" or "better-behaved" than 
. However, the probability that the sequence of flips of a fair coin will wind up in 
is well-defined, for the rather silly reason that the probability is zero (to get into 
, the coin must come up tails on every even-numbered flip).
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a Polish spacePolish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
is universally measurable if it is measurable with respect to every completeComplete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
on
that measures all BorelBorel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
subsets of
. In particular, a universally measurable set of realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s is necessarily Lebesgue measurable (see #Finiteness condition) below.
Every analytic set
Analytic set
In descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .- Definition :There are several equivalent definitions of analytic set...
is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.
Finiteness condition
The condition that the measure be a probability measureProbability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
; that is, that the measure of
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, N0=
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.
Example contrasting with Lebesgue measurability
Suppose is a subset of Cantor spaceCantor space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space...
; that is, is a set of infinite sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of zeroes and ones. By putting a binary point before such a sequence, the sequence can be viewed as a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
between 0 and 1 (inclusive), with some unimportant ambiguity. Thus we can think of
as a subset of the interval Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
. That value is sometimes called the coin-flipping measure of
, because it is the probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
of producing a sequence of heads and tails that is an element of
, upon flipping a fair coin infinitely many times.Now it follows from the axiom of choice that there are some such
without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an , the probability that the sequence of flips of a fair coin will wind up in is not well-defined. This is a pathological property of that says that is "very complicated" or "ill-behaved".From such a set
, form a new set by performing the following operation on each sequence in : Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Now is intuitively no "simpler" or "better-behaved" than . However, the probability that the sequence of flips of a fair coin will wind up in is well-defined, for the rather silly reason that the probability is zero (to get into , the coin must come up tails on every even-numbered flip).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
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.