Maharam’s Theorem
June 27th, 2005 by WaltI’ve been reading Fremlin’s book, and I’ve seen a very surprising theorem that was new to me: Maharam’s Theorem. If you take an set of coins, you can define a measure space on the set of coin flips by taking the product measure. This is a probability measure: the measure of a set is the probability of a coin flip appearing in that set. Since it is a probability measure, it’s well-defined for sets of every cardinality.
You can combine any two measure spaces by taking their disjoint union; the measures are combined by addition. More generally, you can take a weighted sum. Maharam’s theorem states that every nontrivial complete measure space can be constructed from sets of coin flips by taking weighted sums. For example, counting measure is an infinite sum of flips of a single coin. Lebesgue measure on the unit interval arises from flipping an infinite number of coins.
This means that there are not very many types of (complete) measure spaces.