Terence Tao, An Introduction to Measure Theory, AMS GSM 126, 2011. The MATH5825 companion text. Tao develops the theory partly through the exercises themselves — Lebesgue measure from elementary measure, the integration convergence theorems, modes of convergence, differentiation theorems, product measures — with the exercises woven inline through the text rather than collected at section ends.
Solutions to every exercise live at Solutions to Tao’s An Introduction to Measure Theory.