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Toggle## Measure-probability theories: Video lectures

We shall present measure, integral and probability theories, which are very useful in many fields. In particular, they give solid basis not only for statistics theory and machine learning, but also for information theory, queueing theory and point processes, …

### I Measure and integral theories

**PDF**, **Video**, **Vidéo**Measure theory builds solid mathematical foundations for integration and probability. It also underlies information and queuing theories, as well as stochastic geometry. Furthermore, measure theory is very useful in machine learning frameworks, for instance in cases of probability distributions for objects of infinite dimensions.:

**Outline**:

1 Measurability

Algebra and sigma-algebra

Measurable functions

2 Measures

Extension and uniqueness of measures

Cumulative distribution function

Almost everywhere

Lebesgue measure

3 Lebesgue integral

Exchange of integral and limit

Change of variable

Fubini’s theorem

Integration by parts for measures

Radon-Nikodym theorem

Lebesgue decomposition

Lp spaces

### II Fundamentals of probability theory

**PDF**, **Video**, **Vidéo**

The theory of probability can be viewed as a particular case of measure theory. We shall present some of its results which are useful in many fields in mathematics, physics and informatics. In particular, they are very useful in machine learning (ML) frameworks, for instance to establish performance guarantees of ML algorithms.

**Outline**:

1 Probability space

2 Random variable

Expectation of random variable

Moments and characteristic function

CDF and PDF

Independence

3 Convergence theorems for random variables

Monotone convergence

Dominated convergence