Limits & Indicator Functions

Proved a simple theorem today. Let be a sequence of subsets. Then, we have

Proofs use a tedious amount of notation manipulation, but thinking of it in terms of indicator functions is interesting.

Probability Axiomitazation

So I’ve been thinking about sigma-fields and probability measures. The only trick was trying to figure out how conditional probability works.

In some books, is taken as a definition. The question is how does one make the connection with the initial definitions to a notion of conditional probability without getting mired into defining things such as ‘occurences’, ‘events’ etc.

Basically, we define a new probability measure that is ‘restricted’ to B. If the initial probability measure was defined as P(A) = |A|/|X|, then this restricted measure is defined as .

This gives us the conventional definition of conditional probability for discrete spaces.

Things to do next:

1. Derive Bayes’ theorem.