This course consists of two (at times closely interrelated) parts:
Part I: Measure and integration
Part II: Bounded linear operators on Banach spaces and Hilbert spaces.
Literature (part I): the book "A course in real analysis" by John McDonald and Neil Weiss, 2nd edition. Final syllabus consists of the following sections: 1.4 (up to Prop 1.15), 3.1 (only the statement of Thm 3.4, which we took as definition of Borel sigma-algebra), 3.2, 3.4, 4.1, 4.2, 4.3 (with a slightly different proof of Theorem 4.9 that did not use Lemma 4.4), 4.4, 5.1, 5.2, 5.3, 5.4, product measure (see the lecture notes or section 6.3), 6.4 (Theorems 6.6 and 6.7, see the notes for proofs), 13.1, 13.4 (up to and including Thm 13.11). Section 14.3 (which overlaps with parts of chapters 6 and 7 in Rynne and Youngson).
Literature for part II: the book "Linear Functional Analysis" (2nd edition) by Bryan P. Rynne and Martin A. Youngson, Springer Undergraduate Mathematics Series, ISBN 978-1-84800-004-9. Final syllabus: 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 3.4, 4.1 (except 4.12 and 4.13), 4.2, 4.3, 5.1 (only Thm 5.2), 6.1 (up to and including Lemma 6.11), 6.2, 6.4 (up to and including Corollary 6.53), 7.1, 7.2 (up to and including Corollary 7.23, but excluding Theorem 7.18), 7.3 (up to and including Corollary 7.36).