Material for Preliminary Exam B – Analysis

1. Measure Theory
2. Convergence
3. Integration
4. Product measures
5. Signed measures
6. Real line functions

· Fields, sfields, additive set functions · Measures, Measure extension theorem · salgebras of Borel and Lebesgue sets · LebesgueStieltjes measures on the real line · Measurable functions · Convergence in m (measure) · Convergence almost everywhere (m  a.e.) · Almost uniform convergence – Egorov theorem · Simple functions · Definition of ò f dm and its properties · Fatou Lemma · Monotone convergence theorem · Lebesgue dominated convergence theorem · Uniform Integrability (UI) · Convergence in Lp(m) · Construction of the Product Measure mxn · Fubini theorem · HahnJordan decomposition, Total variation · RadonNikodym theorem · Lebesgue decomposition · Differentiation of monotone functions · Absolute continuity

REFERENCE H.L. Royden, P.M. Fitzpatrick Real Analysis, 4th Edition (or any other source addressing topics 16 with detailed proofs).
