Measure and Integration, Fourier

To acquire a foundation in the theory of measure and integration, and then use that foundation to introduce the spaces and tools of functional analysis.

This lesson will cover the following topics:

- General theory of measure: measurable spaces, measurable functions, and measured spaces.

- General theory of integration: integrals of step functions, positive measurable functions, and then real or complex functions. Monotone and dominated convergence theorems. Continuity and differentiability of integrals depending on a parameter.

- Examples of measures: image measures and the transfer theorem, the counting measure on N, the Lebesgue measure on^{R^n}, product measures, and Fubini's theorem.

-^Lpspaces: Hölder and Minkowski inequalities, definition of^Lp spaces. Convolution product and density theorems for^Lp spaces on^Rn.

- Fourier transform on R: definition and properties, inverse formula, example of use.

The analysis courses in the first and second years, specifically:

- HAX403X Analysis 4: Function Sequences, Integral Series, Fourier Analysis

- HAX404X Topology of^{R^n} and Functions of Several Variables

Recommended prerequisites: L2 Math

Hourly volumes:

            CM: 36

            TD: 36

            Practical Work: -

            Land: -