Spring 2022 at IIT Gandhinagar

Assignments

### Logistics

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MeasureTheoryCoursePlan2022.pdf

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<aside> 🖇️ Logistics

• Lectures: Tuesdays, Wednesdays, Fridays: 10:05AM to 11AM
• Venue: AB 7/104 Link to Microsoft Teams (Code: stf9s7z)
• Office Hours: open-door policy and appointment by email </aside>

### Course Description

In this course, we develop the theory of Lebesgue integral. We do this after a brief discussion of the limitations of the Riemann integral that the students are expected to have already learnt. First, we must discuss the question of assigning a measure to as large a collection of sets as possible. This done, we introduce the notion of a simple function. We then develop an integration theory due to Lebesgue by approximating functions by simple functions instead of step functions. Applications are plenty, we discuss only the theory of \$L_{p}\$ spaces and time permitting, the theory of direct integrals.

### Topics

Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theorem. Measure spaces, Outline of extension of measures from algebras to the generated sigma-algebras, Measurable sets; Lebesgue Measure and its properties. Measurable functions and their properties; Integration and Convergence theorems. Introduction to \$L_{p}\$-spaces, Riesz-Fischer theorem; Riesz Representation theorem for \$L_{2}\$ spaces. Absolute continuity of measures, Radon-Nikodym theorem. Dual of \$L_{p}\$-spaces, Product measure spaces, Fubini's theorem. Fundamental Theorem of Calculus for Lebesgue Integrals.