# Measure theory

Marty Ross

#### Synopsis

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.

This subject will be a relatively standard introduction to measure theory, with some emphasis on geometric aspects and applications. More generally, the subject will be a revision of (or an introduction to) the results and techniques of real analysis. In broad outline, the topics we intend to cover are:

• Infinity: friend or foe?
• Outer measure and measurable sets
• Lebesgue measure, Borel and Radon measures
• Measurable functions and integration
• Convergence theorems and $L^p$ spaces
• Iterated integrals and Fubini's theorem
• Hausdorff measure, covering theorems and the area formula
• Differentiation of measures and the fundamental theorem of calculus

#### Contact hours

28 hours spread over the four weeks, with consultation a requested/required.

#### Prerequisites

We will assume some familiarity with the fundamental concepts of analysis on the real line and in Euclidean Space (infs and sups, convergent sequences, open and closed sets, continuity, completeness and compactness, countability). Corresponding familiarity with these notions in metric spaces would be very helpful but will not be assumed; familiarity with these notions in topological spaces would be dandy.

#### Assessment

Problems assigned during lectures (50%), plus a take home exam (50%).