MAT:P-03 Real analysis and measure theory(e-PGP)

Total Lectures

31

Lebesgue measurable sets and their properties (MAT)

Lectures

Lebesgue measurable sets and their properties (MAT)
Notion of lebesgue measure and its basic properties (MAT)
Lebesgue measurable functions and basic properties (MAT)
Lebesgue measurable functions: almost everywhere concept and its implications (MAT)
Simple functions as building blocks of lebesgue measurable functions (MAT)
Relation between almost everywhere convergence and convergence in measure
Lebesgue integration of bounded measurable functions: Riemann integration and lebesgue integration
Convergence of sequences of measurable functions
Lebesgue integral of non-negative functions and Monotone Convergence Theorem (MAT)
Most general notion of Lebesgue integrability and ominated convergence theorem (MAT)
Functions of bounded variations (MAT)
Functions of bounded variations and associated concepts: Vitali Covering Theorem (MAT)
Functions of bounded variations and associated concepts: absolutely continuous functions (MAT)
Functions of bounded variations and associated concepts: differentiability (MAT)
Some useful results of Lebesgue integration (MAT)
Fundamental Theorem of Integral Calculus for lebesgue integration (MAT)
Abstract measure and abstract outer measure (MAT)
Abstract measure theory: extension of measure and the notion measurable covers (MAT)
Abstract measure theory: complete measure and the notion of completion (MAT)
Riemann-Stieltjes integral and its basic properties (MAT)
Notion of Darbaux Stieltjes integral and its implications (MAT)
Motivation and introduction of the notion of Lebesgue integration of bounded functions (MAT)
Examples and some further observations on measurable sets (MAT)
Lebesgue measure: existence of a non-measurable set (MAT)
Lebesgue measure: characterization of measurable sets and further observations (MAT)
Lebesgue measure: notion of inner measure (MAT)
Lebesgue outer measure: definition and properties (MAT)
Abstract measure theory: rings and σ-rings (MAT)
Lebesgue integration of bounded measurable functions: equivalence (MAT)
Convergence of sequences of measurable functions: almost uniform convergence (MAT)
Abstract measure theory: monotone classes (MAT)
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