To make the study easier and better rounded, just add some material.
Rudin's Principles is really nice, especially in retrospect once understand it, but going in as a student it can seem quite severe. It's precise but not too severe -- he just makes you go a chapter or two before it becomes clear why he is doing what he is doing.
To help, my nutshell view is that mostly he is just trying to develop the Riemann (Stieltjes) integral. His main result is, the Riemann integral exists for continuous functions on compact sets. So, then, he needs to say what a compact set is. Well, the most relevant example is just a closed interval of real numbers such as [a,b]. So, why compact? Because every continuous function on a compact set is uniformly continuous, and that lets us know that the Riemann sums converge. What is compact? Every open cover has a finite subcover, and that lets us get uniform continuity. And, in R^n, a set is compact if and only if it is closed and bounded. So, Rudin needs to talk about closed versus open -- he does that on metric spaces although really he needs it only on R^n.
So, net, he starts with metric spaces and discusses open, closed, and compact. Then he shows that in R^n, compact is the same as closed and bounded. He shows that a continuous function on a compact set is uniformly continuous. Then, presto, he shows that the Riemann (or Riemann-Stieltjes if you wish) sums converge and the Riemann integral exists.
He does some nice work on infinite sequences and series, and the main reason is that he uses those tools to show lots of limits exist, e.g., for sines, cosines, and Fourier series.
There's more of value in Principles, but IMHO I gave you a good start to make the book easier. I wish I'd had been given that outline when I was working through Principles at 1+ hour a page.
But the Lebesgue integral in Royden is the one to take fully seriously.
Make Halmos the second or third text on linear algebra. And then look at some quantum mechanics where they discuss eigenvalues, eigenvectors, Hermitian, unitary, and the spectral decomposition! Right, the Halmos book is baby Hilbert space. Then look at some applied connections, e.g.,
George E.\ Forsythe and
Cleve B.\ Moler,
{\it Computer Solution of Linear Algebraic Systems,\/}
Maybe spend an evening on the documentation of LINPACK.
Some weekend of great fun, take a fast pass through the Gauss, ..., Stokes theorem parts of
Tom M.\ Apostol,
{\it Mathematical Analysis:
A Modern Approach to Advanced Calculus,\/}
where don't take the proofs very seriously but to see how physical science and engineering look at calculus of several variables.
Neveu is a great last probability text but not a good first text. So, before Neveu, look quickly, not very seriously, at whatever, including in some introductory statistics texts.
Also, Breiman's Probability is easier to read than Neveu. So, is K. L. Chung's competitive book. And there are others.
There is some more advanced material, e.g.,
Ioannis Karatzas and Steven E.\ Shreve,
{\it Brownian Motion and Stochastic Calculus,
Second Edition,\/}
Rudin's Principles is really nice, especially in retrospect once understand it, but going in as a student it can seem quite severe. It's precise but not too severe -- he just makes you go a chapter or two before it becomes clear why he is doing what he is doing.
To help, my nutshell view is that mostly he is just trying to develop the Riemann (Stieltjes) integral. His main result is, the Riemann integral exists for continuous functions on compact sets. So, then, he needs to say what a compact set is. Well, the most relevant example is just a closed interval of real numbers such as [a,b]. So, why compact? Because every continuous function on a compact set is uniformly continuous, and that lets us know that the Riemann sums converge. What is compact? Every open cover has a finite subcover, and that lets us get uniform continuity. And, in R^n, a set is compact if and only if it is closed and bounded. So, Rudin needs to talk about closed versus open -- he does that on metric spaces although really he needs it only on R^n.
So, net, he starts with metric spaces and discusses open, closed, and compact. Then he shows that in R^n, compact is the same as closed and bounded. He shows that a continuous function on a compact set is uniformly continuous. Then, presto, he shows that the Riemann (or Riemann-Stieltjes if you wish) sums converge and the Riemann integral exists.
He does some nice work on infinite sequences and series, and the main reason is that he uses those tools to show lots of limits exist, e.g., for sines, cosines, and Fourier series.
There's more of value in Principles, but IMHO I gave you a good start to make the book easier. I wish I'd had been given that outline when I was working through Principles at 1+ hour a page.
But the Lebesgue integral in Royden is the one to take fully seriously.
Make Halmos the second or third text on linear algebra. And then look at some quantum mechanics where they discuss eigenvalues, eigenvectors, Hermitian, unitary, and the spectral decomposition! Right, the Halmos book is baby Hilbert space. Then look at some applied connections, e.g.,
George E.\ Forsythe and Cleve B.\ Moler, {\it Computer Solution of Linear Algebraic Systems,\/}
Maybe spend an evening on the documentation of LINPACK.
Some weekend of great fun, take a fast pass through the Gauss, ..., Stokes theorem parts of
Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/}
where don't take the proofs very seriously but to see how physical science and engineering look at calculus of several variables.
Neveu is a great last probability text but not a good first text. So, before Neveu, look quickly, not very seriously, at whatever, including in some introductory statistics texts.
Also, Breiman's Probability is easier to read than Neveu. So, is K. L. Chung's competitive book. And there are others.
There is some more advanced material, e.g.,
Ioannis Karatzas and Steven E.\ Shreve, {\it Brownian Motion and Stochastic Calculus, Second Edition,\/}
Good luck!