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

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.:

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

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.

1 Probability space
2 Random variable
      Expectation of random variable
      Moments and characteristic function
      CDF and PDF
3 Convergence theorems for random variables
      Monotone convergence
      Dominated convergence

Mohamed Kadhem KARRAY

My research activities at Orange aim to evaluate the performance of communication networks, by combining information, queueing theories, stochastic geometry, as well as machine and deep learning. Recently, I prepared video lectures on "Data science: From multivariate statistics to machine and deep learning" available on my YouTube channel. I also teached at Ecole Normale Supérieure, Ecole Polyetechnique, Ecole Centrale Paris, and prepared several mathematical books.