We will mainly use the following two books:
a) "Topics in Real and Functional Analysis" by G. Teschl; and b) "An Introduction to Measure and Integration" by I. K. Rana. You can download a free copy of Teschl's book at his homepage. More precisely, we will cover Chap. 4, 7, 8, 9, and 10 of Teschl's book together with 7.1-7.4 (product measures, Tonelli-Fubini), 8.2-8.3 (modes of convergence), and 9.1 (conditional expectations) in the book of Rana.
More concise syllabus (pensum):
From Topics in Real and Functional Analysis by G. Teschl,
Chap 7. §§ 7.6, 7.7.
Chap. 9. §§ 9.1, 9.2, 9.3.
Chap. 10.
Chap. 4
2) Another source of material for this course is the book "Real Analysis", by G. B. Folland. This books covers virtually everything we shall do (and much, much more). Of particular interest are
Chap. 2 (§§ 4 and 5) Modes of Convergence, Product Measures.
Chap. 3 (§§ 1-4) Decomposition and differentiation of measures.
Chap. 6 (§§ 1 and 2) L^p duality, Riesz' Representation Theorem.
Chap. 5 (§§ 2 and 3) Banach spaces and the theorems on open mapping, closed graph, uniform boundedness, and Hahn-Banach.