|
|
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? And haven't we already gone through all this with David C.
Ullrich in our company? Failing upon a Transfer Principle or some such.
Han de Bruijn
|
|