Teacher: Pertti Mattila 

Scope: 10 cr

Type: Advanced studies

Teaching: 
Weeks 36-42 and 44-50, Wednesday and Thursday 14-16 in room C122. First lecture Wednesday, September 2.

The last lecture and exercise session will be on December 3.

The exams will be on December 10 and 21.

Topics:

Real analysis II is an advanced course on general measure and integration theory and analysis in Euclidean spaces. It is based on and continues the courses  "Mitta ja integraali" ja "Reaalianalyysi I".

Contents:

• Extension and uniqueness theorems for measures
• Product measures and  Fubini's theorem
• Hausdorff measures and dimension
• Weak topology of measures
• Mass distribution principle and Frostman's lemma
• Besicovitch and Vitali covering theorems
• Differentiation of measures and the Radon-Nikodym theorem
• Rademacher's theorem (differentiability of Lipschitz functions)
• Whitney's covering and extension theorem

Course material: Ilkka Holopainen: Moderni reaalianalyysi (in Finnish)

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• MoRA.pdf

Additional lecture notes:

ra2-lecture_notes.pdf

Books partially related to the course

• A.M. Bruckner, J.B. Bruckner and B. Thomson: Real Analysis (Prentice Hall)
• L. Evans & R. Gariepy: Measure theory and fine properties of functions (CRC Press)
• K. Falconer: Fractal geometry: Mathematical foundations and applications (Wiley & Sons)
• K. Falconer: The Geometry of Fractal Sets (Cambridge University Press)
• F. Jones: Lebesgue integration on Euclidean spaces (Jones and Bartlett)
• P. Mattila: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability (Cambridge University Press)
• W. Rudin: Real and complex analysis (McGraw-Hill)

Prerequisites:
Basics on Lebesgue measure and integral