Mathematical writings by Steve Cheng

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.

1. (taylor) We should write out the induction argument in Remark 4.9. The result is self-evident but the notation messy.
2. (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.)
3. (taylor) Should we discuss more kinds of derivatives? (e.g. in topological vector spaces)
4. (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.)
5. (lebesgue) Finish the bits on complete measures.
6. (lebesgue) Prove associativity of finite product measure.
7. (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.
8. (lebesgue) Write an introduction to geometric measure theory.
9. (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.
10. (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?)
11. (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.
12. (lebesgue) Revise the silly footnote on page 4.
13. (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.

Recent changes

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.

PlanetMath

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

Math and computers

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