Measure Theory

Duration

Four Weeks

Lecturer

Marty Ross (Melbourne)

Consultation Hours:

Tuesdays 16–17h Carslaw Room 821

Assessment

50% problems assigned in lectures and 50% take home exam.

Assumed Knowledge

We will assume familiarity with the fundamental concepts of analysis in Euclidean Space (infimum and supremum, open and closed sets, continuity, completeness and compactness, countability). Some corresponding familiarity with these notions in metric spaces would be helpful but will not be assumed; familiarity with these notions in topological spaces would be just peachy.

Background Material

Lecture notes summarising the relevant background on sets and real analysis (PDF) are available. Some (but definitely not all) of this material will be reviewed along the way, particularly the material on metric spaces and topological spaces. Before the summer school begins, you should definitely take a good look at the background notes and, if need be, browse through a real analysis text or two.

Course Outline

Measure theory is the modern theory of integration, the method of assigning a ”size” to subsets of a universal set. It is more general, more powerful and more beautiful (though also more technical) than the classical theory of Riemann integration. The course will be a reasonably standard introduction to measure theory, with some emphasis upon geometric aspects. We will cover most (but definitely not all) of the topics listed below, subject to time and taste:

• General Measure Theory (Outer measure; Measurable sets; Borel and Radon measures; the Caratheodory criterion for Borel measures)
• Special Measures on Euclidean Space (Lebesgue measure; Hausdorff measure; the Vitali Covering Theorem; Hausdorff dimension)
• Integration (Measurable functions; integration and convergence theorems; the Area Formula; iterated integrals and Fubini’s Theorem)
• Functional Analysis (Measures as linear functionals; L_p spaces; the Riesz Representation Theorem)
• Further Topics (Differentiation of measures; the Besicovitch Covering Theorem; the Generalised Fundamental Theorem of Calculus; the Co-Area Formula)