that measures all Borel subsets of
In particular, a universally measurable set of reals 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.
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.
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.
as a subset of the interval [0,1], and evaluate its Lebesgue measure, if that is defined.
, 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.
, the probability that the sequence of flips of a fair coin will wind up in
: Intersperse a 0 at every even position in the sequence, moving the other bits to make room.
, the probability that the sequence of flips of a fair coin will be in
, the coin must come up tails on every even-numbered flip, which happens with probability zero.
To see that, we can test it against a biased coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.
For a set of sequences to be universally measurable, 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.
is tested by the coin we mentioned (the one that always comes up tails on even-numbered flips, and is fair on odd-numbered flips), the probability to hit