منصة مودل - المدرسة العليا للأساتذة بالقبة

The course Measure and Integration provides a rigorous introduction to  measure theory and measurable functions. It extends the classical concepts of length, area, and integration and establishes the mathematical foundations required for advanced studies in analysis, probability theory, and functional analysis.

The course begins with a review of set theory and set-theoretical limits, then introduces σ-algebras, measurable spaces, and measures. Students study measurable functions, simple functions, approximation techniques, null sets, and several notions of convergence such as convergence almost everywhere, almost uniform convergence, and convergence in measure.

Target Audience:

This course is intended for undergraduate students in mathematics, particularly those specializing in pure or applied mathematics. It is designed for students who already possess foundational knowledge in real analysis, linear algebra, and basic topology.

Learning Objectives:

• Master the fundamental concepts of set theory used in measure theory.
• Understand families of sets, indexed collections, and set-theoretical limits.
• Define and manipulate σ-algebras and measurable spaces.
• Construct and analyze Borel σ-algebras.
• Understand measures and their fundamental properties.
• Work with null sets and almost everywhere concepts.
• Define and study measurable functions.
• Perform operations involving measurable functions.
• Use simple functions to approximate measurable functions.
• Analyze different types of convergence of measurable functions.
• Apply theoretical results to exercises and mathematical proofs.
• Build a solid foundation for further studies in Lebesgue integration, probability theory, and functional analysis.