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Han de Bruijn a écrit :
Denis Feldmann wrote:
Here is what comes closest to HdB "ideas" : take a non standard
integer N ("infinite", ie greater than all standard ones, of course).
Define the measure of a standard set of integers A as p(A)= the sum of
1/N for all n <N in A* (A* is the extension of A to the non-standard
integers) For instance, p("n is even")= 1/2 or 1/2-1/N according as N
is even or odd; and the shadow of p(A), p°(A), is a real having
"almost" the properties of a probability (could it be the density
:-)), while p(A) *is* (of course) a discrete measure... and so
additive. Note that with this definition, p(n=42)=1/N and the shadow
p°(n=42)= 0, according to HdB "intuitions". Note also that the main
point of contention (ie p(A), or p°(A)are nor probabilities (ie not
countably additive)) is somewhat shunted...
Why a "non standard" integer? Why is "large enough for our purpose" not
good enough?
If you need to ask, you will never understand. All this talk breaks down
at infinity
And haven't we already gone through all this with David C.
Ullrich in our company? Failing upon a Transfer Principle or some such.
It only makes impossible the result you want, ie a countably additive
measure. Except you dont want it...
Han de Bruijn
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