Mathematical foundations of probability


Mathematical Statistics, Centre for Mathematical Sciences, Lund University with Lund Institute of Technology

Mon 5 (7.5 ECTS credits)

Tatyana Turova, phone: +46 46 222 8543, office: 220-219, email:

Time period:
Fall 2016, period 2.

Course book:

The course will closely follow the book by A. N. Shiryaev, Probability, Springer 1996 (or any other edition).

30 points (45 ECTS credits) in Mathematics. An introductory course in probability and a second course in probability are desirable but not necessary.

FIRST MEETING:                    November 1,   2016,

                                                       time:     10:15 -12:00

                                                     place:      MH 228

Further schedule:    

Starting November 1, lectures run 3 times a week, 2 hours each :

Tuesday 10-12 in MH:228

Wednesday 8-10 in MH:227

Thursday 10-12 in MH:227

The last lecture is on December 22



January 13, time: 9-12, MH:228

January 16, time: 12-15, MH:227

Home work: Chapter II: read pages: 131-205

Problems: 1: 1-4, 2: 1-3, 3: 1, 2, 5, 8, 9, 4: 1-8, 5: 1, 6: 1-14.




Content  of the course   MASM14

1. $\sigma $-algebras, measures, measurable space.

Theorem on the equivalent conditions for the finitely additive set function to be a


2. Borel $\sigma $-algebras, the measurable spaces (${\mathbb R}$,${\cal B}({\mathbb R}$)),

$({\mathbb R}^n,{\cal B}({\mathbb R}^n) )$ and $({\mathbb R}^{\infty},{\cal B}({\mathbb R}^{\infty}) )$.

3. Probability measures on measurable spaces. Theorem on one-to-one correspondence

between the probabilities and the distributions.

4. Measures: discrete, continuous, singular. An example of a singular measure.

    Kolmogorov's Theorem on the extension of measures in (${\mathbb R^{\infty}},{\cal B}({\mathbb R^{\infty}}$) )

5. Random variables. Lemma: Borel function of a random variable is a random variable.

6. Theorem on the limits of the sequences of the extended random variables.

7. $F_{\xi}$ $\sigma $-algebra. Theorem on the representation of the $F_{\xi}$-measurable

random variable.

8. Random elements. Definition of the independence. The necessary and sufficient

conditions of the independence.

9. Lebesgue integral: definition, properties.

10. Theorem on monotone convergence.

11. Fatou's Lemma.

12. Lebesgue's Theorem on Dominated convergence.

13. Chebyshev's, the Cauchy-Bunyakovskii and Jensen's inequalities.

14. Lyapunov's, Hölder's and Minkowski's inequalities.

15. Theorem on change of variables in a Lebesgue integral.

16. Fubini's theorem.

17. Conditional expectation with respect to a $\sigma $-algebra: definition and properties.

18. Theorem on taking limits under the expectation sign.

19. Characteristic functions. Uniqueness theorem.

20. Helly's Theorem.

21. Prokhorov's Theorem.

22. Continuity Theorem.