Handbook

Measure theory is one of the few theories which permeates all core mathematical domains (pure, applied and statistics). We develop Lebesgue integration and probability theory from the core elements of measure theory. The initial background will be kept to a minimum. In particular, it is only required knowledge of real analysis and elementary probability theory (prior knowledge of functional analysis is not required, but it is definitely encouraged). On the other hand, the topics covered in this course will be fundamental for the understanding of advanced courses (differential geometry, advanced analysis, partial differential equations), as described above.

The unit will cover such pure topics as: semi-rings, algebras, and sigma-algebras of sets, measures, outer measures, the Lebesgue and Borel measures, construction of Vitali sets, construction of non-Borel Lebesgue measurable sets, measurable and integrable functions, the Lebesgue integral and the fundamental theorems, change of variables formula in Euclidean space, the Lebesgue spaces, iterated measures and the Fubini theorem, modes of convergence, signed measures, decomposition of measures and the Radon-Nikodym theorem, approximation results for the Lebesgue measure, Hausdorff measure and dimension, Haar measures, ergodic measures.

The unit will also cover topics which are essential for probability theory: such as Borel-Cantelli Lemma, independence, Kolmogorov 0-1 law, exponential bounds, conditional expectation, martingales.