In the previous talk of this series, Ed Dean gave a brief overview of computability theory, defined Martin-Lof randomness, and proved that there is a universal Martin-Lof test for randomness. This last result specifically shows that almost every sequence is Martin-Lof random. This motivates the following question: What theorems of measure theory and probability that contain the words "almost everywhere" or "almost surely" can be replaced with the stronger clause "for all Martin-Lof randoms"? A simple but informative example is the Strong Law of Large Numbers, which states that for almost every sequence of zeros and ones (in the fair-coin probability measure) the limit of the average of the digits is 1/2. In this talk I will prove this result for Martin-Lof random sequences, as well as prove a few other theorems about Martin-Lof randomness.

This is the the second in a series of talks by Ed Dean and Jason Rute.