HotShot Blog

Today was the first day of class. Technically, classes started last week, but since I missed the first Probability Limit Theorems lecture, today was the first day for me.

I'm auditing two courses this semester: Probability Limit Theorems and a seminar in derivatives. Limit theorems is all the Ph.D. level probability theory you would ever want to learn (and the theory you would never want to learn).

So today I found out that for probability measures, convergence in measure (ie convergence in probability) implies convergence almost everywhere. Of course, for general measure spaces, this is not true.

Consider the example f_n (x) = 1 on [n,n+1] ; 0 otherwise

This function converges to 0 for every x, ie f_n (x) -> 0 as n->infinity. But the integral of f_n is 1 for all n, so f_n does not converge in Lebesgue measure.