Re: Calculus XOR Probability

On Mon, 20 Mar 2006 13:18:40 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

You've also stated that _because_ of this obvious fact
it follows that the sum of infinitely many zeroes is 1.

I'm in the position now to replace that by the correct result, with
Robinson's non-standard analysis, that I didn't know of at the time
I wrote (wrongly) the above. Let *R be set of all hyperreal numbers.

Let e = (1,1/2,1/3,1/4, ... ,1/n, ...) in *R
Let N = (1,2,3,4, ... ,n, ...) in *R

Then it follows that e . N = 1 in *R .

And no, due to the Transfer Principle, you don't have to start from
scratch, if you want to extend Probability Theory to the hyperreals.

Right.

You can certainly do probability theory _in_ nonstandard analysis.
And you can use that to prove things about standard probability
theory. But using nonstandard analysis to prove things about
standard probability theory doesn't _change_ what standard
probability theory _is_. The transfer principle converts
statements about hyperreals into statements about their
standard parts; the standard part of an infinitesmal is 0.

I'm tempted to ask for a precise statement of the transfer
principle. Instead I'll just ask this: Suppose x is a real
number between 0 and 1, chosen at random. Which infinitesmal
gives the probability that x = 1/2?

Seems like you need to be able to answer questions like that
if you're going to be applying nonstandard analysis to probability
in the way it seems you think you can...

Han de Bruijn


************************

David C. Ullrich
.