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

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.