Set Operations and Algebras of Sets
Sequences, limsup, liminf
Open and closed sets of reals
Continuous functions
Borel sets

Lebesgue measure
Outer measure
Measurable sets
Measurable functions

Lebesgue integral
Convergence theorems
Fatou's Lemma
Monotone Convergence Theorem
Lebesgue Dominated Convergence Theorem

Classical Banach Spaces
L^p spaces
Minkowski and Hölder inequalities
Completeness

General Measure and Integration
Measure spaces, measurable functions, convergence theorems
Signed measures, examples of measures
Radon-Nikodym Theorem
Lebesgue decomposition, singular, absolutely continuous
Product measures
Fubini's and Tonelli's Theorems

Modes of Convergence
Definitions of Pointwise (A.E.), Uniform, Mean P, in measure
Counterexamples
Relating various modes (Chapter 7 in Bartle, for example)
Egoroff's Theorem

References:

Real Analysis, H. Royden, Macmillan Publishing Co.
Measure and Integration, M. Munroe, Addison-Wesley Publishing Co.
Elements of Integration, Bartle, Wiley Classic Series
Real and Abstract Analysis, Hewitt and Stromberg.