I'm a graduate school student entered this year.
Next semester, I would take mathematical statistics.
I don't think the book used for this class is rigorous A Course in Mathematical Statistics, Second Edition.
But I want to understand Statistics deeply, rigorously.
I hear often Measure Theory is useful for deep understanding of Mathematical Statistics.
So I have studied Real Analysis. G Folland from chapter 1 measure to chapter 3 Radon-Nikodym Theorem and I will study some more.
Could you give me a advice?
I am familiar with this book. Your current background and planned reading in real analysis should be sufficient for navigating measure theory at the level used in the book. There is some use of characteristic functions (Fourier transforms), which require a bit of complex analysis.
This is a second edition. In the preface to the first edition Roussas mentions linear algebra and advanced calculus as adequate prerequisites.
This book provides a solid, rigorous and very traditional approach to mathematical statistics. Do not look for simulation or for any Bayesian approaches.
I suggest you read prefaces to first and second editions (both included) and browse the first few chapters as you begin the course. That will give you more information than I can include here, and I suspect nothing to discourage you.