Material for Preliminary Exam B --- Analysis Section  

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 books suggested below.

 

1.      Measure Theory

·        Fields, s-Fields, Additive Set Functions

·        Measures, Measure Extension Theorem

·        s-Algebras of Borel and Lebesque Sets

·        Construction of non-Measurable Set

·        Lebesgue - Stieltjes Measures on R

2.      Convergence

·        Measurable Functions

·        Convergence in m (measure)

·        Convergence Almost Everywhere (m - a.e. )

·        Almost Uniform Convergence - Egorov Theorem

3.      Integration

·        Simple Functions

·        Definition of  ∫ f dm and Properties

·        Monotone Convergence, Fatou Lemma,

Lebesque Dominated Convergence Theorem

·        Convergence in L^p(m)

4.      Signed Measures

·        Hahn-Jordan Decomposition

·        Radon-Nikodym Theorem/Applications

·        Lebesque Decomposition Theorem

5.      Product Measures 

·         Construction of the Product Measure mxn

·         Fubini Theorem

 

REFERENCE

 

H.L. Royden, Real Analysis (or any other source addressing topics 1-5 with detailed proofs).