Measurement and integration, Fourier

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

This EU will address the following:

- General measurement theory: measurable spaces, measurable applications and measured spaces.

- General theory of integration: integrals of stepped functions, positive measurable functions and real or complex functions. Theorems of monotonic and dominated convergence. Continuity and derivability of integrals depending on a parameter.

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

- ^Lp spaces: 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, Series of functions, integer series, Fourier

- HAX404X Topology of ^Rn and functions of several variables

Recommended prerequisites: L2 math

Hourly volumes:

            CM : 36

            TD : 36

            TP: -

            Land: -