I am a master's statistics student who is just about to finish studying Terence Tao's Analysis I and Analysis II books and I really liked his way of teaching things. Besides its content, I also need to study measure theory for researching purposes. Based on my needs and background, I would like to know if it is a good idea to study Terence Tao's An Introduction to Measure Theory or should I reinforce my theoretical basis before doing so.
If this is the case, I am very keen to study the topics covered by Rudin's Principles of Mathematical Analysis in order to complement what I have learned so far, but I would like to know first if there is a similar book which covers the same topics and is written in the same or similar way as Tao's does.
I am also interested in any book recommendation related to measure theory, but once again I'd prefer texts written similarly to the way Terence Tao's does.
The answer to your question is YES: Terence Tao's Analysis I and II provide you with all of the necessary knowledge to complete a graduate measure theory course. You do not need to go over other books.
Having said that, it could help you a lot to have a good intuition in topology when dealing with Lebesgue measures. Also, there are some extra difficult exercises, e.g. Exercise 1.1.11 would require some knowledge of abstract linear algebra and linear isomorphisms, or in Exercise 1.1.17 knowledge of polyhedral geometry is kind of necessary; but you can complete the course without having to solve these.
Finally, if you are still having difficulties, you can consult solution manuals here.