Any advise and suggestions about Real analysis and measure theory?

Question

I am about to take a real analysis course and i wanna ask if anyone can provide some good texts or reference or any other source. The lectuer indeed suggested the Rudin real and complex analysis but i wonder if there are more and better texts and sources. The course content are provided as follows.Thx

Course content:

Lebesgue Measure on $\mathbb{R}$: Measurable sets and Lebesgue measure, Measurable functions

The Lebesgue Integral: The Lebesgue integral, modes of convergence

Differentiation and Integration: Functions of bounded variation, Differentiation of an integral, absolute continuity

General Measure and Integration Theory: Measurable spaces, measurable functions, integration, convergence theorems, the Radon-Nikodym theorem

The $L^p$ Spaces: The $L^p$ spaces, convergence and completeness, bounded linear functionals

Answer 1

I personally really like Folland's Real Analysis, which will cover all of the topics you mention. There is also a good text by Halmos but you presentation and notation is a bit old-fashioned. If you are looking for a more recent account there is Tao's book on measure theory that will cover the first 3.5 topics (I believe Radon-Nikodym is not covered). Tao's book is very concrete though, which you might find an advantage or disadvantage according to the precise style of your lecturer and your own comfort with the abstract.

Answer 2

I find the Zakon series good, complete and thorough. The way he organizes his sections in logical progression is very elegant.

Answer 3

I personally studied on Rudin's Real and complex analysis, which is rather abstract and synthetic. But I also like Royden's Real analysis, which treats Lebesgue's measure on $\mathbb{R}$ and abstract measure theory. A more recent book is Charles Pugh's, that I recommend for a first study.

Answer 4

For the part about Lebesgue measure on the real line, I can recommend Neal L. Carothers's Real Analysis.