I am an average student and have to study following topics on my own for the exam :
The measure of a bounded interval in $\mathbb R^n$ , the Riemann integral of a bounded function defined on a compact interval in $\mathbb R^n$ , Sets of measure zero and Lebesgue’s criterion for existence of a multiple Riemann Integral, Evaluation of a multiple integral by iterated integration.
Please can anyone suggest some good self-study book providing good insight into the above topics ..
On
There is an excellent presentation of the Riemann-Stieljies integral in several dimensions in Tom Apostol's classic Mathematical Analysis. Several other good sources for this material are Harold Edwards' Advanced Calculus of Several Variables, Rosenlicht's Introduction to Analysis and John and Barbara Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Hubbard and Hubbard is a true classic;a beautifully written modern presentation of rigorous calculus of several variables using linear algebra and differential forms. It's loaded with historical notes,examples and beautiful computer generated graphs you won't find anywhere else. If you can afford it, I would heartily recommend it above all others. Edwards is similar to- but not quite as comprehensive as-Hubbard and Hubbard in spirit and coverage and has the advantage of being cheap.Sadly, many of the physical applications are shunted to the exercises,which is the only real flaw in the book.Rosenlicht is cheap,concise and wonderfully clear-and it has one of the best presentations of the Riemann integral in both one and several variables there is. All have very clear and excellent accounts of the multivariable Riemann integral.
A bit more difficult but also a very important source is Analysis on Manifolds by James Munkres. Munkres' book requires a stronger background, some analysis in metric spaces and a good working knowledge of linear algebra. But it has one of the best rigorous presentations of multivariable calculus there is,including the Reimann integral.
As far as measure theory goes, I would avoid it and the Lebesgue integral at this level. I think jumping directly into the Lebesgue integral is too demanding for most mathematics majors until they've mastered a strong analysis course like Pugh or Rudin.
I found the second edition of Elementary Classical Analysis by Marsden to be a gentle and geometrically motivated guide to these topics. The topics you mention are covered in Chapters 8 and 9. The book is also good for studying since it contains many problems for each chapter, with half of them (the odd ones) having solutions in the back. It's a bit lacking in measure theory since it mainly talks about sets of measure zero for Lebesgue's theorem. I don't know how far you want to go in measure theory - for example if you want to see the existence of non-measureable sets. If you do want to go further into measure theory, which naturally develops into the study of Lebesgue integration, I would suggest the fourth edition of Royden.