Material for Preliminary Exam B – Analysis


The following list gives topics on which the Preliminary Examination B in Analysis will be based. Math 5317 covers many, but not necessarily all, of these topics. Students (even those who have taken Math 5317) are advised to prepare for the examination using many resources, including (but not limited to) the book suggested below.

 

 

1.    Measure Theory

 

 

 

 

2.    Convergence

 

 

 

 

3.    Integration

 

 

 

 

 

 

 

 

 

4.    Product measures

 

 

5.    Signed measures

 

 

 

6.    Real line functions

 

·         Fields, s-fields, additive set functions

·         Measures, Measure extension theorem

·         s-algebras of Borel and Lebesgue sets

·         Lebesgue-Stieltjes 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

·           Hahn-Jordan decomposition, Total variation

·           Radon-Nikodym 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 1-6 with detailed proofs).