Measurement and integration, Fourier

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

This UE will cover the following points:

- 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. Monotone and dominated convergence theorems. Continuity and derivability of integrals dependent 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.

- ^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.

Analysis courses in L1 and L2, in particular :

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

- HAX404X Topology of ^Rn and functions of several variables

Recommended prerequisites: L2 maths

Hourly volumes :

            CM: 36

            TD : 36

            TP: -

            Land: -