I would like to teach myself measure theory and I am looking for a good introductory textbook at the advanced undergraduate or early graduate level. A reader-friendly text with plenty of examples would be ideal.
I am a big fan of Bartle's Elements of Integration. You will also want to see a nice development in the special case of Euclidean Space. For this I recommend Wheeden and Zygmund's book, Measure and Integral.
Do like young Grothendieck, try to rebuild Lebesgue theory yourself.
If you fail, Bartle is a good read, and Rudin is not bad either.
I highly recommend Royden's real analysis book. The latest version has a large errata but is well broken down in bite sized chunks for self learning. I know this from experience. The older editions have a smaller errata but also require u to fill in more details. Others may disagree with me here but the worst book for self learning Measure theory would be the big Rudin. The concepts are not reinforced enough for newbies. You could allways read both. I would suggest work through Royden and then do exercises only from Rudin's book. All the best.
I would recommend Measures, Integrals and Martingales by Schilling. It starts out very basic with the notion of $\sigma$-algebras, measures and measurable mappings and also covers integration theory including the most commonly used covergence theorems. Further topics such as martingales, Radon-Nikodym's theorem and conditional expectations are also treated.