Universally Measurable Set - Example Contrasting With Lebesgue Measurability

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, 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.