Measure Theory, Probability, and Stochastic Processes

Measure-Probability: Measure Theory, Probability and Stochastic Processes concepts.

Course Overview

This page brings together graduate-level lectures and exercises on measure theory, probability, and stochastic processes. The material follows a unified measure-theoretic approach, structured in three parts: Part I — Measure Theory and Integration, Part II — Probability and Random Sequences, and Part III — Stochastic Processes and Ergodic Theory. Each lecture is available as a downloadable PDF, accompanied by exercises with detailed solutions. Selected lectures also include video recordings in English and French. This foundational framework underpins the resources available on Statistics Theory, Machine & Deep Learning, Information Theory, Queueing Theory, and Point Processes.

Part I: Measure Theory and Integration

1 Measure Theory

PDF, video, and exercises of the lecture on Measure Theory, the cornerstone of modern integration and probability theory.
📄 PDF 🎬 Video (EN) 🎬 Vidéo (FR) ✏️ Exercises
We begin with the fundamental concept of measurability, exploring algebras and σ-algebras, the construction of measurable sets and generated σ-algebras, and the pivotal Borel σ-algebra. We then study measurable functions, their properties such as measurability at the limit, the concept of simple measurable functions, and the indispensable simple approximation theorem. We move on to a rigorous examination of measures, starting with their definition and essential lemmas. Through insightful examples including the Dirac measure and the weighted counting measure, we elucidate the concepts of finite, probability, and σ-finite measures. We then address the continuity of measures, the extension and uniqueness of measures from algebras and semirings, the cumulative distribution functions, the concept of almost everywhere, and conclude with the existence and uniqueness of the Lebesgue measure on ℝⁿ.

Course Outline:

  1. Measurability
    1. Algebra and σ-algebra
    2. Measurable functions
  2. Measures
    1. Extension and uniqueness of measures
    2. Cumulative distribution function
    3. Almost everywhere
    4. Lebesgue measure

2 Lebesgue Integral

PDF, video, and exercises of the lecture on the Lebesgue Integral, a profound extension of the classical Riemann integral.
📄 PDF 🎬 Video (EN) 🎬 Vidéo (FR) ✏️ Exercises
We begin by detailing the construction of the Lebesgue integral, with essential results on the exchange of integral and limit operations, exemplified by the Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem. We present the Change of Variable formula, providing a rigorous framework for transforming integrals between spaces. We then focus on Fubini’s theorem, investigating the existence and uniqueness of product measures and the Fubini-Tonelli theorem, which provides conditions under which the order of integration can be interchanged. We explore the Radon-Nikodym theorem, central to understanding the relationship between measures and functions, with the notions of absolute continuity and the Lebesgue decomposition. Finally, we turn to Lp spaces, covering essentially bounded functions, Minkowski’s inequality, Hölder’s inequality, the celebrated Riesz-Fischer theorem, the theory of the Hilbert space L², and convergence properties within Lp spaces.

Course Outline:

  1. Construction of the Lebesgue integral
    1. Exchange of integral and limit
    2. Change of variable
  2. Fubini’s theorem
    1. Integration by parts for measures
  3. Radon-Nikodym theorem
    1. Lebesgue decomposition
  4. Lp spaces

3 Integration of Functions of a Real Variable

PDF, videos and exercises of the lecture on Integration of Functions of a Real Variable, adapting to the Lebesgue framework classical results that are traditionally presented for the Riemann integral.
📄 PDF 🎬 Video (EN) 🎬 Vidéo (FR) ✏️ Exercises
After constructing the Lebesgue integral on general measure spaces in the previous lectures, this chapter restricts to mappings of a real variable, with ℝ equipped with the Lebesgue measure. Many classical results of integration on ℝ are traditionally stated for the Riemann integral, as in the well-known textbook Cours de Mathématiques Spéciales by Ramis, Deschamps and Odoux. We adapt them here to the Lebesgue framework, for mappings with values in ℝ̄n or ℂn. We develop the Chasles relation, integral mapping and antiderivatives, the mean value formulas, the Newton-Leibniz formula, integration by parts, change of variables, Taylor’s formula with integral remainder, integrability by comparison, the integration of comparison relations, and finally the generalized integral with its calculation techniques. A pedagogical highlight is that integration is a regularizing operation: starting from a possibly discontinuous integrand, the integral mapping is always continuous, and differentiable almost everywhere thanks to the Lebesgue Differentiation Theorem.

Course Outline:

  1. Introduction
    1. Motivation: from Riemann to Lebesgue
    2. Oriented Integral
  2. Properties of the Integral on ℝ
    1. Chasles Relation
    2. Integral Mapping and Regularity
    3. Antiderivatives
    4. Mean Value Formulas
  3. Integral vs. Antiderivatives
    1. Newton-Leibniz Formula
    2. Integration by Parts
    3. Change of Variables
    4. Taylor’s Formula with Integral Remainder
  4. Integrability by Comparison
    1. Comparison by Inequality
    2. Domination and Equivalence
  5. Integration of Comparison Relations
    1. Integrable Case
    2. Non-Integrable Case
    3. Reference Functions and Comparison Rules
    4. Bertrand Integrals
  6. Generalized Integral
    1. Link with the Lebesgue Integral
    2. Correspondence with Riemann Terminology
    3. Cauchy Criterion for Integrals
    4. Abel’s Rule and Oscillating Integrals
    5. Bilateral Generalized Integral
  7. Calculation Techniques for a Generalized Integral
    1. Generalized Newton-Leibniz
    2. Change of Variables for the Generalized Integral
    3. Generalized Integration by Parts

Part II: Probability and Random Sequences

4 Probability Theory

PDF, video, and exercises of the lecture on Probability Theory, developed from its measure-theoretic foundations.
📄 PDF 🎬 Video (EN) 🎬 Vidéo (FR) ✏️ Exercises
We begin by defining the probability space, encompassing the probability measure, events, and the concept of « almost surely » within probability theory. We then explore random variables and their associated probability distributions, covering the definition and characteristics of random variables, the σ-algebra generated by them, and the expectation of random variables with its key properties. We discuss moments, characteristic functions, cumulative distribution functions (CDFs), and probability density functions (PDFs), along with the notion of independence among random variables. Finally, we examine the monotone and dominated convergence theorems for random variables, providing crucial insights into the behavior of sequences of random variables.

Course Outline:

  1. Probability Space
  2. Random Variable
    1. Expectation of Random Variable
    2. Moments and Characteristic Function
    3. CDF and PDF
    4. Independence
  3. Monotone and Dominated Convergence for Random Variables
    1. Monotone Convergence for Random Variables
    2. Dominated Convergence for Random Variables

5 Discrete Random Variables and their Transform

PDF and exercises of the lecture on Discrete Random Variables and Their Transform.
📄 PDF ✏️ Exercises
We begin with the fundamental types of discrete random variables: the Bernoulli random variable, representing the simplest form of a random process with two outcomes; the Binomial random variable, modeling the number of successes in independent trials; the Geometric random variable, focusing on the number of trials needed to achieve the first success; and the Poisson random variable, modeling the number of events occurring in a fixed interval. We then explore the probability generating function (PGF), starting with its definition and fundamental properties, illustrating how it encapsulates the probabilistic information of a discrete random variable. We discuss how the PGF can be used to analyze the distribution of a random variable, derive its factorial moments, handle random sums of random variables, and exploit its monotonicity and convexity properties.

Course Outline:

  1. Fundamental discrete random variables
    1. Bernoulli random variable
    2. Binomial random variable
    3. Geometric random variable
    4. Poisson random variable
  2. Probability generating function
    1. Definition and properties
    2. Examples of probability generating functions
    3. Probability generating function and distribution analysis
    4. Factorial moments from probability generating function
    5. Random sum of random variables
    6. Monotonicity and convexity of the probability generating function

6 Continuous Random Variables and their Transforms

PDF and exercises of the lecture on Continuous Random Variables and Their Transforms.
📄 PDF ✏️ Exercises
We begin with the fundamental continuous random variables, focusing on the Gaussian random variable, renowned for its bell curve and central role in the Central Limit Theorem, and the Exponential random variable, crucial in modeling time until an event. We then explore the moment generating function (MGF), with its formal definition, the conditions for its existence through the radius of convergence, and its infinite series expansion. We cover the characteristic function, a Fourier transform of the probability density function, discussing its differentiability, the implications of its finite expansion, the pivotal Lévy’s inversion formula, and its infinite expansion. Finally, we discuss the Laplace transform, particularly useful in solving differential equations and analyzing systems, covering its differentiability, finite expansion, and infinite expansion, and noting that the Laplace transform is real analytic.

Course Outline:

  1. Fundamental continuous random variables
    1. Gaussian random variable
    2. Exponential random variable
  2. Moment generating function
    1. Definition and radius of convergence
    2. Infinite expansion of the moment generating function
  3. Characteristic function
    1. Differentiability and finite expansion of characteristic function
    2. Lévy’s inversion formula
    3. Characteristic function of random vectors
    4. Infinite expansion of the characteristic function
  4. Laplace transform
    1. Differentiability and finite expansion of Laplace transform
    2. Infinite expansion of the Laplace transform
    3. Laplace Transform is Real Analytic

7 Random Vectors and Gaussian Distribution

PDF and exercises of the lecture on Random Vectors and Gaussian Distribution.
📄 PDF ✏️ Exercises
We begin with the mathematical backbone of random vectors through the lens of characteristic functions, starting with Lévy’s inversion formula for random vectors, pivotal for recovering the distribution of random vectors from their characteristic functions, and the independence criterion, which provides a method to determine the statistical independence of components within random vectors. We then focus on square-integrable random vectors, introducing the covariance matrix, a critical tool for understanding the variance and correlation structure of random vectors, covering degenerate random vectors and the role of affine transformations. We turn to Gaussian random vectors, beginning with their definition and characteristic function, examining the relationship between independence and uncorrelation, and analyzing the probability density function of a nondegenerate Gaussian vector. Finally, we explore complex Gaussian random vectors, starting with the basics of complex random vectors and progressing to the properties of symmetric complex Gaussian vectors, the criterion for the independence of jointly CN random vectors, the spectral decomposition and characterization of CN vectors, the real representations of complex vectors and matrices, the covariance and linear combinations of CN random vectors, and the probability density function of CN random vectors.

Course Outline:

  1. Characteristic Functions of Random Vectors
    1. Lévy’s Inversion Formula for Random Vectors
    2. Independence Criterion
  2. Square-Integrable Random Vectors
    1. Covariance Matrix
    2. Degenerate Random Vectors
    3. Affine Transformations
  3. Gaussian Random Vectors
    1. Definition and Characteristic Function
    2. Independence Versus Uncorrelation
    3. Probability Density Function of a Nondegenerate Gaussian Vector
  4. Complex Gaussian Random Vectors
    1. Basics of Complex Random Vectors
    2. Symmetric Complex Gaussian Vector
    3. Criterion for Independence of Jointly CN Random Vectors
    4. Spectral Decomposition and Characterization of CN Vectors
    5. Real Representations of Complex Vectors and Matrices
    6. Covariance and Linear Combinations of CN Random Vectors
    7. Probability Density Function of CN Random Vectors

8 Convergence for Sequences of Random Variables

PDF and exercises of the lecture on Convergence for Sequences of Random Variables.
📄 PDF ✏️ Exercises
We begin with almost-sure convergence, covering the Strong Law of Large Numbers, the Borel-Cantelli lemmas, and various conditions guaranteeing almost-sure convergence. We then focus on convergence in probability, a weaker form of convergence, with the Continuous Mapping Theorem for convergence in probability. We discuss convergence in quadratic mean (L² convergence), particularly important in the context of mean-square error analysis and signal processing. We progress to weak convergence (convergence in distribution), covering the Portmanteau theorem, the Continuous Mapping Theorem for weak convergence, the Characteristic Function Criterion, and the fundamental Central Limit Theorem. We synthesize the different types of convergence through the connections between convergence types, explore the concept of stochastic order for comparing the magnitudes of random variables, and address the concept of tightness of probability measures with the pivotal Prohorov’s Theorem.

Course Outline:

  1. Almost-sure convergence
    1. Strong law of large numbers
    2. Borel-Cantelli lemmas
    3. Conditions for almost-sure convergence
  2. Convergence in probability
    1. Continuous mapping theorem for convergence in probability
  3. Convergence in quadratic mean
  4. Weak convergence – Convergence in distribution
    1. Portmanteau theorem
    2. Continuous mapping theorem for weak convergence
    3. Characteristic function criterion
    4. Central limit theorem
  5. Connections between convergence types
  6. Stochastic order
  7. Tightness of probability measures
    1. Tightness and uniform tightness
    2. Lemmas on Tightness
    3. Prohorov’s Theorem

Part III: Stochastic Processes and Ergodic Theory

9 Stochastic Processes

PDF and exercises of the lecture on Stochastic Processes, mathematical models depicting the evolution of systems over time through probabilistic mechanisms.
📄 PDF ✏️ Exercises
We begin by defining stochastic processes and discussing the concept of independence within these processes. This foundation sets the stage for understanding finite-dimensional distributions (fidis), essential for grasping the behavior of processes across various dimensions. We delve into Kolmogorov’s Extension Theorem, which provides a method for constructing a process from its finite-dimensional distributions, and we analyze sample paths, exploring their properties and behaviors, before concluding with second-order stochastic processes, focusing on their mean and covariance functions. We then differentiate between strict stationarity, which requires the joint distribution of any set of points to be invariant under time shifts, and wide-sense stationarity, which relaxes this to only require invariance in the mean and autocovariance functions. We explore the measurability of stochastic processes and introduce the stochastic integral, a critical tool in stochastic calculus. Finally, we cover Gaussian processes, starting with their definition and characteristic functions, the conditions for their existence and stationarity properties, and conclude with an in-depth look at the Wiener process, also known as Brownian motion, a cornerstone of stochastic modeling in finance and physics.

Course Outline:

  1. Foundations of Stochastic Processes
    1. Definition and Independence of Stochastic Processes
    2. Finite-Dimensional Distributions (fidis)
    3. Kolmogorov’s Extension Theorem
    4. Sample Paths of a Stochastic Process
    5. Second-Order Stochastic Process
  2. Strict and Wide-Sense Stationarity
    1. Stationary Stochastic Processes
    2. Wide-Sense Stationarity
  3. Measurability and Stochastic Integral
    1. Measurable Stochastic Processes
    2. Stochastic Integral
  4. Gaussian

10 Conditional Expectation and MMSE Estimation

PDF and exercises of the lecture on Conditional Expectation and MMSE Estimation, two fundamental concepts in probability theory and statistical estimation.
📄 PDF ✏️ Exercises
We begin with conditional expectation, starting with the fundamentals of conditioning with respect to a sigma-algebra and a random variable. We explore its properties and the important convergence theorems (monotone and dominated convergence) in the context of conditional expectation. We distinguish between conditional expectations for discrete and continuous variables and examine the special case of conditional expectation in Gaussian distributions. We then focus on Minimum Mean Square Error (MMSE) estimation, highlighting its relationship with conditional expectation, comparing MMSE estimation to conditional expectation and emphasizing their similarities. We finally introduce linear MMSE estimation, discussing its practical applications and advantages in various scenarios such as statistical analysis and signal processing.

Course Outline:

  1. Conditional Expectation
    1. Conditioning With Respect to a Sigma-Algebra
    2. Conditioning With Respect to a Random Variable
    3. Properties of Conditional Expectation
    4. Monotone and Dominated Convergence for Conditional Expectation
    5. Conditional Expectation for Discrete and Continuous Variables
    6. Conditional Expectation in Gaussian Case
  2. MMSE Estimation
    1. MMSE Estimation vs. Conditional Expectation
    2. Linear MMSE Estimation

11 Martingales and Ergodic Theory

PDF of the lecture on Martingales and Ergodic Theory, two pivotal areas in probability and dynamical systems.
📄 PDF
We begin with an introduction to martingales, focusing on their definition and basic properties, highlighting their significance in stochastic processes. We delve into the optional stopping theorem and martingale convergence theorems, crucial for understanding the behavior of martingales under various conditions, and examine several examples of martingales illustrating their application in different probabilistic contexts. We then turn to ergodic theory, starting with the notions of flow, shift, and compatibility, essential for understanding the dynamics of systems over time. We introduce the stationary framework, providing a foundation for analyzing systems that exhibit statistical regularity. A key highlight is Birkhoff’s Pointwise Ergodic Theorem, a fundamental result that connects individual trajectories with long-term statistical behavior. Finally, we explore the ergodic stationary framework, offering insights into the behavior of systems that are both stationary and ergodic.

Course Outline:

  1. Martingales
    1. Definition and Basic Properties of Martingales
    2. Optional Stopping and Martingale Convergence Theorems
    3. Examples of Martingales
  2. Ergodic Theory
    1. Flow, Shift and Compatibility
    2. Stationary Framework
    3. Birkhoff’s Pointwise Ergodic Theorem
    4. Ergodic Stationary Framework

Book Coming Soon

A book based on this material is currently in preparation: Mohamed Kadhem KARRAY — « Measure Theory, Probability, and Stochastic Processes: An Integrated Approach ». The book will provide a unified treatment of measure theory, probability, and stochastic processes, expanding and consolidating the lectures available on this page. Stay tuned for the publication announcement.

About These Topics

These graduate-level lectures on measure theory, probability, and stochastic processes are designed for students and researchers seeking a rigorous mathematical foundation for advanced applications in statistics, machine learning, information theory, queueing theory, and stochastic geometry. The material adopts a unified measure-theoretic approach, where probability theory is naturally derived from measure theory, providing a coherent framework across the three parts. Part I covers the foundations of measure theory, the construction of the Lebesgue integral, key results such as Fubini’s theorem, the Radon-Nikodym theorem, and Lp spaces, and the classical theory of integration on the real line. Part II develops probability theory from its measure-theoretic foundations, covering random variables and their transforms (probability generating functions, moment generating functions, characteristic functions, Laplace transforms), random vectors and Gaussian distributions, and the various modes of convergence of sequences of random variables including the Central Limit Theorem and Prohorov’s theorem. Part III introduces stochastic processes, covering foundations through Kolmogorov’s extension theorem, stationarity, the stochastic integral, Gaussian processes, the Wiener process, conditional expectation and MMSE estimation, martingales, and ergodic theory including Birkhoff’s pointwise ergodic theorem. Each lecture is accompanied by exercises with detailed solutions, designed to reinforce the theoretical material and build problem-solving skills. The material draws on classical references in measure theory and probability, presented in a self-contained format suitable for independent study, graduate coursework, or research preparation.

Last Updated on 29 mai 2026 by Mohamed Kadhem KARRAY