I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying theories in more detail, since these mostly include Measure Theory, Lebesgue Measures and Lebesgue Integration, due to their probabilistic nature.
My problem is, I was mostly educated as a "coder" (Bs and Msc in Computer Science as well), not as a mathematician. Unfortunately I am still relying on my sloppy background which I obtained in the superficial Calculus courses I had taken years ago. Clearly, I am in need of building at least a working knowledge about Real Analysis, which covers topics like Lebesgue Theory, Multivariable Calculus, rigorous definitions of limits, derivation, integration etc. Since I am little bit late to study all of these, I am not able to spend my little available time with books having hundreds and hundreds of pages, it wouldn't be possible.
So, I am in need of an advice on a good and brief (as possible) Real Analysis book, which has the beginners as its target audience, as possible, so I can study it chapter by chapter from start to finish and can build myself a working knowledge of Lebesgue Integration, Measure Theory,etc. I read somewhere that Rudin's book is good but is too much detailed. By the way, I am also open to any advice on how to study Real Analysis in an efficient way, by myself.
Thanks in advance.