We shall indicate two different sources to the course contents:
1) The book "Real Analysis", by G. B. Folland will cover virtually everything we shall need (and much, much more). Of particular interest to us 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.
2) "Topics in Real and Functional Analysis" by G. Teschl will also cover the course, when supplemented by "An Introduction to Measure and Integration" by I. K. Rana. In particular,
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
should be sufficient.
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