Bressoud, David M., A Radical Approach to Lebesgue’s Theory of Integration (Cambridge: Cambridge University Press, 2008), MAA textbooks <http://www.loc.gov/catdir/enhancements/fy0803/2007035326-t.html>
Capi*nski, Marek, and P. E. Kopp, Measure, Integral and Probability, 2nd ed (London: Springer, 2004), Springer undergraduate mathematics series
Rudin, Walter, Real and Complex Analysis, 3rd ed (New York: McGraw-Hill, 1987), McGraw-Hill series in higher mathematics
Tao, Terence, An Introduction to Measure Theory (Providence, R.I.: American Mathematical Society), Graduate studies in mathematics