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 Lp(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).