Measurement and integration, Fourier

Acquire the fundamentals of measure theory and integration, then use these fundamentals to introduce the spaces and tools of functional analysis.

This EU will address the following points:

- General theory of measure: measurable spaces, measurable applications, 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, counting measures on N, Lebesgue measures on^Rn, 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, inversion formula, example of use.

The L1 and L2 analysis courses, in particular:

- HAX403X Analysis 4, Function sequences, entire series, Fourier

- HAX404X Topology of^Rn and functions of several variables

Recommended prerequisites: L2 maths

Hourly volumes:

            CM: 36

            TD: 36

            TP: -

            Land: -