Measure and Integration I (MAA5616)
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.
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.