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## Re: [probability] Re: does projection preserves measurability?

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• Noel, First of all, thank you very much for your answer. It is proving very useful. Second, I am not as aquainted with measure theory as I would like and my
Message 1 of 16 , Feb 3, 2005
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Noel,

First of all, thank you very much for your answer. It is proving very
useful.
Second, I am not as aquainted with measure theory as I would like and my
knowledge on it is scattered, so:
Third, I'll keep picking your brain if you do not mind.

>
> I hope to come back to you vey soon with a counter-example.
>

Don't mind about the counterexample, it was only to understand better.

> There is a result I know about the product B(R+)xW, where
> (W,F,P) is some probability space and B(R+) is the
> Borel sigma-algebra on R+. The space R+xW is obviously
> very important for the study of stochastic processes.
>
> The following theorem can be found in:
> Claude Dellacherie, Paul-Andr� Meyer
> Probabilit�s et Potentiel, Ch I a IV,
> Hermann 1975
> Theoreme III.44 page 103.

I checked on this book and on the theorem, and it is actually standing on
Theorem III.13 (page 43, in my english copy) which states:

Theorem
(1) B(R) \subset A(K(R)) , A(B(R)) = A(K(R))
(2) Let (W,F) be a measurable space. The product \sigma-field G=B(R)xF
on RxW is contained in A(K(R)xF).
(3) The projection onto W of an element of G (or, more generally, of A(G))
is F-analyitic.

K(R) a is the paving of R consisitng of all compact subsets of R,
an A(G) is the paving on RxW consisting of all G-analyitc sets.

Dellacherie and Meyer stated this theorem for R being the reals. Item (3)
(which is the one I am interested in) is proved by Pollard in a note I
found a couple of days ago
(http://www.stat.yale.edu/~pollard/603.fall04/notes/analytic-sets.pdf)
but taking R to be any compact metric space.

My question is (and here I may be either silly or naive): if in addition
F is an analytic space, wouldn't then be the projection a measurable set
on F?

And an aside question: what is the difference between analytic sets
(spaces) and Souslin sets (spaces)? I have seen this names used sometimes
interchangably.

Thanks, again,
Pedro

PS: I will start reading the D&M's book, I just hope not to feel too
frustrated (so far I have only had an interrupted reading up to tutorial 4
of you www.probability.net, which, by the way, I think is beautiful and
great for self-learning)

--

Pedro R. D'Argenio dargenio@...

FaMAF - Universidad Nacional de Cordoba Tel: +54 351 4334051 (365)
Ciudad Universitaria Fax: +54 351 4334054
(5000) Cordoba
Argentina http://www.cs.famaf.unc.edu.ar/dargenio/
• ... Pedro, I am very sorry, but your questions have gone beyond my current state of knowledge so I am not able to answer them. I did make the effort of reading
Message 2 of 16 , Feb 4, 2005
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> My question is (and here I may be either silly or naive):
> if in addition F is an analytic space, wouldn't then be
> the projection a measurable set on F?

> And an aside question: what is the difference between
> analytic sets (spaces) and Souslin sets (spaces)? I have
> seen this names used sometimes interchangably.

Pedro,

I am very sorry, but your questions have gone beyond
my current state of knowledge so I am not able to answer
them. I did make the effort of reading D&M (I-IV) line
by line about 10 years ago, (which is why I remembered
the fact that measurability of projections was dealt
with in some cases), but I have forgotten the concepts
of "analytic set/space" and "Souslin set/space". When
I retire from banking and work full time in mathematics
I ll be able to help you more, promised :-)

> PS: I will start reading the D&M's book, I just hope not
> to feel too frustrated

I think the book is very difficult and requires some
maturity in the subject. It was a mistake for me to
read it as a beginner. In general, with the insight
of more experience, I think it a mistake to read books
which are too difficult line by line. It is better to
read them only to get some intuition and flavour, and
wait for the day when your brain is better prepared
to really attack them.

> so far I have only had an interrupted reading up to
> tutorial 4 of you www.probability.net, which, by the
> way, I think is beautiful and great for self-learning)

Thank you Pedro

Noel.
• Noel ... Thanks anyway, Noel. Despite I still do no t have the answer, I feel relieve that I am not asking naive questions in the matter. ... Well, I it seems
Message 3 of 16 , Feb 4, 2005
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Noel

> I am very sorry, but your questions have gone beyond
> my current state of knowledge so I am not able to answer

Thanks anyway, Noel. Despite I still do no t have the answer, I feel
relieve that I am not asking naive questions in the matter.

> When
> I retire from banking and work full time in mathematics
> I ll be able to help you more, promised :-)

Well, I it seems a good decision to wait to be a self-funding
do-what-you-want researcher :-) ... Specially in mathematics!!

Thanks again for your help,
Pedro.

--

Pedro R. D'Argenio dargenio@...

FaMAF - Universidad Nacional de Cordoba Tel: +54 351 4334051 (365)
Ciudad Universitaria Fax: +54 351 4334054
(5000) Cordoba
Argentina http://www.cs.famaf.unc.edu.ar/dargenio/
• ... A university professor is not exactly self-funding , but does appear, for the most part, to be a do-what-you-want researcher. Just thought I d point
Message 4 of 16 , Feb 4, 2005
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> Well, I it seems a good decision to wait to be a self-funding
> do-what-you-want researcher :-) ... Specially in mathematics!!

A university professor is not exactly "self-funding", but does
appear, for the most part, to be a "do-what-you-want" researcher.
Just thought I'd point that out in case anyone was consider career
options.

You mentioned "Souslin" spaces. The last time I read that term was
in the book "Radon Measures on Arbitrary Topological Spaces and
Cylindrical Measures" by Laurent Schwartz. I recently tried to buy a
copy of that book, but my web search turned up nothing! There don't
appear to be any copies anywhere. Does anybody here know where I can
find a copy? Or alternatively, can anybody here suggest a different
book that covers the same theory?

This is important stuff in probability. When we impose the Radon
property to a measure, we have many more tools at our disposal. But
the theory of Radon measures, as it is normally developed in a basic
analysis course, is only applicable to measures on a locally compact
Hausdorff space. But in probability, we often want to work with
function spaces which are not locally compact.

For example, I was playing around with white noise some time back.
As such, my probability space was the space of Schwartz
distributions. (White noise can be thought of as the derivative of
Brownian motion, which -- since BM is not differentiable in the
ordinary sense -- lives in the space of Schwartz distributions.) I
wanted to show that a certain class of Schwartz distributions was
dense in my probability space. It would have been the perfect place
to apply to Stone-Weierstrass approximation theorem. But the only
version of that theorem I had at my disposal was the one for Radon
measures on a locally compact Hausdorff space. And the space of
Schwartz distributions is not locally compact.

Anyway, if anyone has any suggestions for me, I would love to hear
them.

Thanks,
Jason
• ... You re very welcome. Noel
Message 5 of 16 , Feb 4, 2005
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> Thanks again for your help,

You re very welcome.

Noel
• Hi Jason, welcome back. ... I can t find any French version either. I have checked his cours d analyse (in 4 volumes) and I can t see anything related to
Message 6 of 16 , Feb 4, 2005
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Hi Jason, welcome back.

> "Radon Measures on Arbitrary Topological Spaces
> and Cylindrical Measures" by Laurent Schwartz.

I can't find any French version either. I have checked
his "cours d'analyse" (in 4 volumes) and I can't see
anything related to radon measures, that goes beyond
the scope of locally compact Hausdorff. In fact, I
have checked all my other (prominent) measure theory
book :-) Can't see anything more general.

> But in probability, we often want to work with
> function spaces which are not locally compact.
> [...] my probability space was the space of Schwartz
> distributions. (White noise can be thought of as the
> derivative of Brownian motion

Obviously this is too advanced for me to be of any
help, but it seems to me you want to play with
functional analysis and probability, with special
emphasis on the Wiener measure. I have three books
which should have a lot of relevant information
(they are among the books I promise myself to
read first when I retire :-)

Probability in Banach Spaces
http://www.amazon.co.uk/exec/obidos/ASIN/0387520139/

Transformation of Measure on Wiener Space
http://www.amazon.co.uk/exec/obidos/ASIN/3540664556/

Stochastic Analysis
http://www.amazon.co.uk/exec/obidos/ASIN/0387570241/

I have also 2 books from Dinculeanu, but probably
less relevant if you stick to probability, i.e.
measures with value in [0,1]

Vector Measures
http://www.amazon.co.uk/exec/obidos/ASIN/0080121926/

Vector Integration and Stochastic Integration In
Banach spaces
http://www.amazon.co.uk/exec/obidos/ASIN/0471377384/

I am sorry I can't do more than throwing a few radom
references at you :-(

--
Noel
• ... 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
Message 7 of 16 , Feb 5, 2005
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--- 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

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
• Hi Zaeem, ... I have ordered Pollard and Aliprantis & Border, which I do not own. Thank you. I am very fond of the He-Wang-Yan as the book seems to be a direct
Message 8 of 16 , Feb 5, 2005
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Hi Zaeem,

> 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).

I have ordered Pollard and Aliprantis & Border,
which I do not own. Thank you.

I am very fond of the He-Wang-Yan as the book seems
to be a direct heir of Dellacherie & Meyer, while
at the same being far easier to read.

Noel.
• Thanks for the references, Noel. There s a text with the same title as one of yours, Vector Measures , by Diestel and Uhl (if I ve spelled those right), which
Message 9 of 16 , Feb 6, 2005
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Thanks for the references, Noel. There's a text with the same title
as one of yours, "Vector Measures", by Diestel and Uhl (if I've
spelled those right), which is pretty relevant stuff, I think. I
don't believe I'll have trouble ordering that one and it's on my
list.

In this day and age, it's hard to believe that it is *impossible* to
get a copy of any particular book. Especially for a mathematician,
or any scientist for that matter. One would think a system would be
in place. (Okay, I guess it is. It's called the "library". But
selfish me wants to own it!) :)
• Hi Jason, ... Yep, I ve got it too :-) Forgot to mention it. ... The main problem is the cost. If one is prepared to pay, there is a very good site to find
Message 10 of 16 , Feb 6, 2005
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Hi Jason,

> Thanks for the references, Noel. There's a text
> with the same title as one of yours, "Vector Measures",
> by Diestel and Uhl (if I've spelled those right),

Yep, I ve got it too :-) Forgot to mention it.

> In this day and age, it's hard to believe that
> it is *impossible* to get a copy of any particular book.

The main problem is the cost. If one is prepared to pay,
there is a very good site to find just anything:

www.abebooks.com

Noel.
• ... Zaaem, I have received the book today. It looks amazing. Thank you Noel.
Message 11 of 16 , Feb 10, 2005
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> Aliprantis and Border, Infinite dimensional
> analysis (2nd ed.) section 10.5, 10.6 (my favourite).

Zaaem,

I have received the book today. It looks amazing.

Thank you

Noel.
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