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 CoArea Formula)
