Title of the course: Measure and Paradoxes in Probability
Instructor: Dr. Tatiana González Grandón
Institution: Humboldt Universität Berlin
Dates: 7-13 August 2017
Prerequisites: Calculus. familiarity with undergraduate-level probability theory is advantageous.
Level: Undergraduate (all levels), graduate
Abstract: In this 6 days course we will introduce students to probability theory with the enticing philosophical
underpinning of human beings wanting to measure uncertainty. For that, we first need to define what measure is?
What does it mean for a subset E of R to be measurable? This will lead us to compare the differences between
measure and cardinality. We will look at some beautiful but non-pathological subsets, which cannot be measured
and perplex us amid the Banach-Tarski Paradox. We will construct the basic ideas and tools to work with the
Lebesgue Integral, while drawing its parallels to Expected values of uncertain results. Moreover, in the deterministic
world it is unambiguous what it means for a sequence to converge to a limit, if however one has a sequence of Random
Variables, how can it converge?
1. The problem of measure: Cardinality, Banach-Tarski Paradox.
2. Monty Hall Problem. What is Probability?
Probability space and examples.
3. Birthday problem.
Random variables as measurable functions.
4. Bertrand’s box paradox.
Expected Values and Lebesgue Integral.
5. Playing with infinity
Types of Convergence and Examples.
6. Distributions of random variables.
Probability and Random Processes, Grimmett Geoffrey.
An Introduction to Measure Theory, Terence Tao.