The following are some notes I made for my own studies. But they are free, so you may find them a useful alternative or supplement to the standard textbooks, or as an introduction to the topic before reading more advanced works.

- Lebesgue Integral and Measure Theory
- Taylor polynomials in Banach spaces
- Conditional probabilities and expectations
- An Application of the Rational Canonical Form
- Formal statement of the method of characteristics to solve linear first-order partial differential equations
- Differentiation under the integral sign using weak derivatives
- Sources for my mathematical writings and diagrams

Warning: there may be mistakes in these notes. Some of them, and other omissions, are listed below. I should get around to fixing them eventually.

- (taylor) We should write out the induction argument in Remark 4.9. The result is self-evident but the notation messy.
- (taylor) In remark 4.5, do not mention the function $J$. (The section that mentions $J$ was later moved, and I forgot to revise the comment.)
- (taylor) Should we discuss more kinds of derivatives? (e.g. in topological vector spaces)
- (lebesgue) I think we should make clear the distinction between $\R$ and $[-\infty, +\infty]$ after all. (Because I plan to write a section on signed measures, and signed measures are not allowed to take infinite values.)
- (lebesgue) Finish the bits on complete measures.
- (lebesgue) Prove associativity of finite product measure.
- (lebesgue) Replace the whole section on vector-valued integrals with a more general treatment of integration in Banach spaces. Ideally I want to have enough machinery to define stochastic integrals.
- (lebesgue) Write an introduction to geometric measure theory.
- (lebesgue) Can we write a nice treatment on the Riesz Representation Theorem? The proof in Folland is fairly messy; I do not know if I can improve on it.
- (lebesgue) The section on construction of Lebesgue measure could use some diagrams. So does the section on Stieljes measure (where did I get those funny estimates?)
- (lebesgue) The third last paragraph of the proof of Theorem 9.1 has a small mistake. It assumes that if $A_i$ covers $C$, then $\closure{A_i}$ covers $\closure{C}$. This is patently false. But this problem is easy to fix, just expand each $A_i$ a little bit beforehand.
- (lebesgue) Revise the silly footnote on page 4.
- (lebesgue) Perhaps we should use $\cM$ instead of $\cA$ to denote sigma-algebras. The notation right now leads to a little confusion when algebras are involved.

- 2007-07-29
- I am now undertaking a rewrite and expansion of the measure theory book. As it will take a while to finish; I am making a draft available.
- 2006-11-30
- Added article on differentiation under the integral sign
- 2006-09-02
- (cond. prob.) Added an example application in Proposition 10.4.
- 2006-08-18
- (cond. prob.) Fixed a mistake in the statement of a theorem. Some proofs had employed the incorrect version of that theorem, and these were fixed too.
- 2006-08-16
- (cond. prob.) Fixed some typos, and added a few more intuitive explanations.
- 2006-06-30
- Added summary on method of characteristics
- 2006-02-16
- (taylor) Fixed the error in the proof that the Taylor polynomial is $o(\norm{x-a}^n)$. Fixed a typo in the product rule corollary.

I am also a contributor to the PlanetMath encyclopedia of mathematics. Some of my other mathematical expositions may be found there.

- My page on TeX
- MathML (did you know that Firefox 1.5 can display MathML? This rocks!)