Real Analysis

Royden is breathtakingly gorgeous. But, it can be good to skip some parts. The main content is just 'measure theory', and that's just freshman calculus grown up. Why? Because near 1900 it became clear that freshman calculus was clumsy for some important progress, especially cases of convergence of functions. Measure theory? Well, first cut, 'measure' is just a grown up version of ordinary area. Simple. Why interested in measure theory? Because want to cook up a new way to do integration, that is, what freshman calculus dues with the Riemann integral. Recall, the Riemann integral partitions the X axis and approximates the area under the curve with tall, thin rectangles. Measure theory partitions the Y axis: At first glance this seems a little clumsy, but in the usual cases get the same number for area under a curve and in bizarre cases, that can get from converging functions, get a nice answer that Riemann integration can't do. Royden likes Littlewood's three principles, and they are cute. So, spend an evening on them. Yes, it's possible to use Littlewood's to do the subject, but there is a better way, also heavily in Royden, roughly called 'monotone…

"Can you please suggest on-line resource to learn great beginning and advanced probability concepts?" Yup! The subject at the level you ask is a major topic in applied math but is not very popular in US universities. So, you will not get many good answers. In particular, the 'computer science' community, with 'machine learning' and 'artificial' this, Bayesian that, likely won't have good answers. Your "on-line" part is asking a bit much; I can give you references to books but not all on-line. There may be some PDF files on-line, from TeX, that have such material; try some Google searching with the keywords used here. The intuitive foundations of probability went back to gambling. 'Probability' is a field of 'applied' math and as such is well defined: About 100 years ago, E. Borel student H. Lebesgue invented 'measure theory' which essentially 'rewrote' classic calculus, especially the part about integration. For the simple cases, what Lebesgue did gets the same numerical values as the classic Riemann integral. The difference is that in theoretical work Lebesgue's integral is much more general and much better 'behaved'.…