2011/12

lectures given by Francesco Leonetti

Measures: definition and examples (Lebesgue, Dirac, pitch and toss).
Measurable sets and their properties.
Sets with zero measure and measurability.
Every set is measurable with respect to Dirac measure and also with respect to pitch and toss measure.
Sigma algebras. Borel sets.
Borel measures and Radon measures.
s-dimensional Hausdorff measure in R^n: definition and properties.
Hausdorff dimension.
A set that is not H^1 measurable.
Cantor set in R^2 with parameter k: how to build it and estimates for its Hausdorff measure H^s.
Bounded functions on sets with finite measure: simple functions (properties and integral), integrable functions (properties and integral)
Integration with respect to the Dirac measure.
Measurable functions, continuous functions and their relation with integrable functions.
Integration for (possibly) unbounded functions on set of (possibly) infinite measure.
Passing to the limit under the integral sign.
Product measure and Fubini's theorem.
Derivative of a measure with respect to another measure. Radon-Nikodym's theorem; Lebesgue's decomposition.
L^p spaces and reflexivity. Linear and continuous functionals on L^p.

References

Lecture notes on measure theory (2002/03) written by Federica Pezzotti

Lecture notes on integration (2003/04) written by Lucilla Macchiagodena ; errata corrige: see here

Lecture notes about Cantor set (2006/07) written by Fabio Aiello and Romina Eramo; errata corrige: see here

Evans L. C. - Gariepy R. F. "Measure theory and fine properties of functions" CRC Press 1992

Rudin W. "Analisi reale e complessa" Boringhieri 1974

Brezis H. "Analisi funzionale" Liguori Editore 1986