--- In

probability@yahoogroups.com, "Noel Vaillant" <vaillant@p...>

Dear all,

I just read the sequence of your posts about analytic sets.

Dellacherie and Meyer is perhaps the first and most widely referenced

source on the application to probability, of Choquet's capacity

theory. The treatment is long and detailed, and typically French in

flavour. There are several, more recent references.

Like Noel, I must admit that this material is not fresh in my mind,

but I will try to provide some intuition:

Perhaps you already know that these results are used, among other

things, to prove measurability of hitting times of subsets of the

product space $\Omega \time S$, where $\Omega$ is a sample space, $S$

is the path space of a stochastic process.

The rough idea is based on the following observation: Continuous

images (images under continuous functinos) of open, (closed),

[measurable] sets are not necessarily open (closed), [measurable]. But

continuous images of COMPACT sets are compact. This suggests that for

the product space $X \times Y$ it may be possible to define a class of

measurable rectangles $A \times B \in X \times Y$ (where $A \in X$ and

$B \in Y$ and $A$ is COMPACT), that "generate" a class of sets whose

projection (an open map) onto $X$ remains measurable.

Here are some references:

Pollard's book A user's guide to measure theoretic probability has

already been mentioned.

Doob's classic - Classical potential theory and its probabilistic

counterpart had a self contained appendix that deals with analytic sets.

Sheng-Wu He, Jia-Gang Wang, Jia-An Yan - Semimartingale Theory and

Stochastic Calculus is has a very readable section on

"section-theorems" which deals with these issues beautifully.

Aliprantis and Border, Infinite dimensional analysis (2nd ed.) section

10.5, 10.6 (my favourite).

There may be slight and subtle differences in the terminology used by

different authors, so its probably best to choose one reference, and

stick with it. For this reason, I prefer the last reference on my list

- the book contains excellent sections on measure theory and general

topology which can be refered to without the danger of inconsistent

terminology.

Best regards,

Zaeem Burq.

> I am sorry I can't do more than throwing a few radom

> references at you :-(

>

> --

> Noel