Expository articles on Analysis and Probability theory

Solution 1:

I think that these notes by John Erdmann might satisfy your requirements for at least a portion of the analysis material. First, there is:

A Problem Text in Advanced Calculus

This set of notes covers basis aspects of topology, basic aspects of integral/differential calculus and concludes with the inverse and implicit function theorems.

There is also:

A Companion to Real Analysis

These notes could perhaps be considered as a continuation of his problem text. More topology is discussed, Lebesgue integration, Banach/Hilbert spaces and concludes with some operator theory. There is also some probability thrown in for good measure, pun intended.

For multivariable analysis/calculus, I also like Jerry Shurman's notes that you can find here. He covers some of the same ground that Erdman does, but goes much farther into "vector calculus" including differential forms and Stokes/Gauss' theorems. Like Erdman's, these notes are also extremely well-written.

Also, speaking of professors named Conrad, you might want to look at Brian Conrad's notes on differential geometry found here. Although these notes cover more DG than you may be interested in, he gives a full accounting of Stokes' theorem and the algebra/analysis that is required to get there. The treatment is considerably deeper than that found in Shurman's notes. Each topic is contained in a separate document and there are probably about 50 or so of these documents, usually between 10 and 20 pages apiece.