The plan is to cover the following topics:

1. Probability spaces
Basic concepts, fields, sigma-fields, Borel sigma-fields, probability measures, subjective probability

2. Random variables and distribution functions
Random variables, simple random variables, continuity of probability measures, Borel lemma, induced probability measures, distribution functions

3. Expectations
Definitions, properties, monotone convergence theorem, Riemann-Stieltjes integral and expectations, inequalities

4. Measure theory
Finite and sigma-finite measures, extension of probability measures, existence and uniqueness

5. Lebesgue integration
Definition of Lebesgue integral, properties, Fatou lemma, convergence theorems (monotone, dominated and bounded convergence), uniform integrability, absolute continuity, Radon-Nikodym theorem

6. Product measures
Product sigma-fields, product probability spaces (finite and infinite), Fubini theorem

7. Independence
Definitions, properties, Borel-Cantelli lemma, Kolmogorov zero-one law

8. Convergence of sequences of random variables
Modes of convergence, relations

9. Random infinite series of independent random variables
Convergence results, Kolmogorov three-series theorem

10. Weak and strong laws of large numbers

11. Conditional probability and conditional expectation
Definitions, properties, Jensen inequality for conditional expectations, convergence theorems for conditional expectations

12. Martingales
Definitions, examples, basic properties, martingale convergence theorems

If time allows, and given the development of the necessary background from the above topics, we will discuss briefly Markov chains, in particular the theoretical results (e.g., the ergodic theorem) that form the basis of modern Bayesian statistical work.