Universally measurable set
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 reals is necessarily Lebesgue measurable (see #Finiteness condition) below.
The condition that the measure be a probability measure; 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.
Example contrasting with Lebesgue measurability
Suppose is a subset of Cantor space ; that is, is a set of infinite sequences 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.