Suggested references
There is no prescribed text, but the following textbooks are suggested:
 Gerald B. Folland, Real Analysis: Modern Techniques and their Applications, Wiley, 1999
 Walter Rudin, Real and Complex Analysis, McGrawHill, 1986
Topics
The PhD examination covers both abstract integration theory and basic
functional analysis. The precise content varies with instructor, but
will include the following:

Measure: σalgebras, construction of measures (the Caratheodory
and HahnKolmogorov theorems), Lebesgue measure on \(\mathbb{R}^n\), product
measures, signed measures and the HahnJordan decomposition 
Integration: measurable functions, the Lebesgue integral, modes of
convergence, the Vitali convergence theorem, the FubiniTonelli
theorem, the RadonNikodym theorem 
Functional Analysis: normed vector spaces and Banach spaces, bounded
linear operators, the HahnBanach theorem and its corollaries, dual
spaces, the Baire Category theorem and its applications (Open Mapping
theorem, Closed Graph theorem, Banach isomorphism theorem), the
BanachSteinhaus theorem, Hilbert spaces, \(L^p\) spaces.
As noted above, additional topics may appear, according to time and
taste; these have included the Lebesgue Differentiation Theorem,
BanachMazur distance between normed spaces, and basic results in
Fourier analysis and probability. Students preparing to take the
examination should consult the most recent instructor of the course
for additional information.
(posted April 9, 2014)