Mathematics
MA490: Measure Theory (5 ECTS)
A "measure" on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. Measure is a generalization of the concepts of length, area, and volume. An important example is Lebesgue measure, which assigns the conventional length, area and volume of Euclidean geometry to suitable subsets of n-dimensional space. Measure Theory is the basis for Integration and it is the foundation for an understanding of Probability Theory.
Taught in Semester(s) 1. Examined in Semester(s) 1.
Workload: 102 hours (24 Lecture hours, 12 Tutorial hours, 66 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Carry out basic operations on sequences of sets.
- Decide whether a given set function is a measure and execute basic operation with measures.
- Apply the theory of integration in a wide range of settings, including the real numbers and probability spaces. Decide when term by term integration of a sequence or series of functions is permissible.
- Give an account of the basic facts about measure spaces and integration.
- Compose and write proofs of theorems about measures and integrals.
Indicative Content
Algebras and $\sigma$-algebras of sets. Measures. Lebesgue Measure. Upper and lower limits for sequences of sets. Probability Spaces. The Borel-Cantelli Lemmas.
Measurable functions and random variables. Modes of convergence for sequences of measurable functions. Approximation by simple functions. The "almost everywhere" concept. Egoroff's Theorem.
Integration of measurable functions. The Lebesgue integral. Integration of random variables. Fatou's Lemma and the Lebesgue Convergence Theorems.
Properties of Lebesgue measure on Euclidean spaces. The Cantor Set. The existence of non-measurable sets.
Module Resources
- "Real Analysis", H.L. Royden, Pearson.
- "Real Variables", A. Torchinsky, Addison Wesley
- "The Elements of Integration and Lebesgue Measure", R. Bartle, Wiley-Interscience
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