Measure and Integration II (MAA5617)
The main text is Folland, Real analysis. For useful examples, see Gelbaum and Olmsted, Counterexamples in analysis. For the connections with category theory, consult Oxtoby, Measure and category. Some standard sources on geometric measure theory include Mattila, Geometry of sets and measures in Euclidean spaces and Evans-Gariepy, Measure theory and fine properties of functions.
Another great book to check out is Kolmogorov and Fomin, Elements of the theory of functions and functional analysis, available as copyleft! A comprehensive reference on measure theory, also available as copyleft, is the treatise of Fremlin.