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.