Notes for the instructor (i.e., me).

Initially, we will review discrete probability, so that we make sure we speake common language. We will supplement this with *interesting* examples, of probability on discrete sets with some "life" in them.

Immediately, we will review (and motivate) measure theory for probabilty, go quickly through the foundations and, through case studies, explain why studying probability or any related field cannot be done without a solid measure-theoretic understanding.

Our motto then will be that
probability = measure theory + independence + coupling
What this means can only be explained through little experience.

Convergence of random series, in the very classical Kolmogorov manner. Strong law of large numbers and Glivenko-Cantelli. Applications: Monte Carlo, Bernstein, etc.

(Quick review of metric space topology with emphasis on compactness.) The structure of the space probability measures. Characterization of compactness of sets of probability measures. Harmonic analysis and Levy's theorem.

Specialize to familiar metric spaces: the circle, the Euclidean space, the space of continuous functions. Construct natural probability measures. (Prove CLT.) Introduce Brownian motion.

Random walks and harmonic analysis will be our next venture to the contribution of probability to mathematics and vice versa. A little understanding of physics is a bit helpful here. Proof of transience in dimension 3 and higher and recurrence in dimension 2.

Markov chains in countable state space. Stationarity and stability. First passage time problems. Perron-Frobenius and other standard results. Specialize to random walks on graphs and explain connections with electric circuits.

There will be a final lecture to wrap up and review things.