NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (4/9)

Advanced Set Theory 

•    Equivalent Sets

•    Countable and Uncountable Sets

•    The concept of a cardinal number

•    The cardinals Ν_º and c

•    Addition and multiplication of cardinals

•    Cartesian product, Axiom of Choice, Multiplication of cardinal numbers

•    Order relation and order types, Well ordered sets, Transfinite induction 

•    Addition and multiplication of ordinals

•    Statements of Zorn’s lemma, Maximality principle and their simple implications



Section-II (5/9)

Measure Theory 

•    Outer measure, Lebesgue Measure, Measureable Sets and Lebesgue measure, Non measurable sets, Measureable functions

The Lebesgue Integral

•    The Rieman Integral, The Lebesgue integral of a bounded function

•    The general Lebesgue integral

General Measure and Integration

•    Measure spaces, Measureable functions, Integration, General convergence theorems

•     Signed measures, The Lp-spaces, Outer measure and measurability

•     The extension theorem

•     The Lebesgue Stieltjes integral, Product measures