Probability in an infinite sample space - sci.math

Message from discussion Probability in an infinite sample space

Robert Low

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More options Mar 29 2006, 9:43 am

Newsgroups: sci.math

From: Robert Low <mtx...@coventry.ac.uk>

Date: Wed, 29 Mar 2006 08:43:24 +0100

Local: Wed, Mar 29 2006 9:43 am

Subject: Re: Probability in an infinite sample space

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Han de Bruijn wrote:
> Robert Low wrote:
>> Bunch of stuff deleted.
> Thank you very much, Robert, for scrutinizing this material. I think we
> can conclude that my sloppy approach can indeed be replaced by something
> better.

Sometimes (and this is the case with the modern approach to
infinitesimals) the way to make it better is to find a
framework in which the kind of argument you want to make
isn't sloppy after all. Of course, then there's the
hard work involved in setting up the framework, and in
figuring out just which apparently sloppy arguments work.

But there are several different ways of approaching what
you want to do in this particular case. After posting
what I did, I realised that the following two
different looking versions are just different aspects
of view of the same underlying argument.

Version 1: using the vector field r-hat / (4 pi r^2) (i.e.
the unit vector pointing radially outward, magnitude
1/(4 pi r^2). The component of this perpendicular to any
surface is the light intensity as seen by the surface.

Version 2: take the uniform area 2-form with total area
1 on the sphere, and map (half) the sphere to the image plane
by taking each point X on the sphere to the point in the image
plane P where the line from the origin through X meets P. Carry
the area form on the sphere to one on the image plane. Then
the induced area form also gives the light intensity as seen by
the surface.

Version 1 requires a dot product: version 2 needs you
to work out an induced map on two-forms. But they're
'really' the same, because the normal to the sphere
is dual to the area form.

Or you could just work with infinitesimal surface elements,
which is essentially version 2 where you're working in
a region sufficiently small that errors involved in assuming
that the surface element is actually plane are negligible.
(The trick, of course, is knowing that there is a 'sufficiently
small', and not just blithely assuming it.)

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