I’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.

That interpretation of Maharam’s theorem is not quite correct. The theorem says that all atomless measure algebras are of this form, where a measure algebra is obtained by taking the quotient with respect to null sets and the notion of isomorphism is a boolean sigma-isomorphism. To see the difference, we can construct a probability space with a measure algebra equilvalent to Lebesgue measure on the unit interval but larger cardinality. Let K be a cardinal larger than [0,1]. Let X=K x [0,1]. We can interpret X as a space partitioned into slices of the form {(k,r):r fixed}. The set of slices corresponds to [0,1]. If B is a Borel set in [0,1], we let B_X={K x B}. The sets of the form $B_X$ form a sigma-algebra identical to the usual Borel sigma-algebra but with points replaced by copies of K. We define a measure mu_X on them by letting mu_X(B_X)=mu(B). Then the constructed measure space is equivalent to the unit interval with Lebesgue measure as a measure algebra but has a larger cardinality. Taking the completion doesn’t change the measure algebra, but the space will contain many more null sets than there are Lebesgue measurable sets in [0,1].

Thanks for the correction.