, a subset
of a Polish space
is universally measurable
if it is measurable
with respect to every complete probability measure
that measures all Borel
. In particular, a universally measurable set of reals
is necessarily Lebesgue measurable
(see #Finiteness condition
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.
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
=[-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
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
, 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.