Probability with Martingales

Introductory remarks

Topology is about open sets. The characterizing property of a continuous function f is that the inverse image f −1(G) of an open set G is open.

Measure theory is about measurable sets. The characterizing property of a measurable function f is that the inverse image f −1(A) of any measurable set is measurable.

In topology, one axiomatizes the notion of ‘open set’, insisting in particular that the union of any collection of open sets is open, and that the intersection of a finite collection of open sets is open.

In measure theory, one axiomatizes the notion of ‘measurable set’, insisting that the union of a countable collection of measurable sets is measurable, and that the intersection of a countable collection of measurable sets is also measurable. Also, the complement of a measurable set must be measurable, and the whole space must be measurable. Thus the measurable sets form a σ-algebra, a structure stable (or ‘closed’) under countably many set operations. Without the insistence that ‘only countably many operations are allowed’, measure theory would be self-contradictory – a point lost on certain philosophers of probability.