In addition to anonymoushn's correct point, I would also observe that even if we do treat infinity as a specific real number, the integration of a function zero at everywhere except one specific point from -inf to +inf is still 0. And further if we are treating it as a real number, you've now violated the uniformity constraint; P(0) != P(inf), therefore it is no longer a uniform distribution, the thing you are putatively building.
For an analysis of this and some other interesting things, see the Dirac delta (psuedo-)function: http://en.wikipedia.org/wiki/Dirac_delta_function
Re:Zerocoin: making Bitcoin anonymous
by anonymoushn on Friday April 12, 2013 @12:41AM
(#5536670)
Infinity isn't a positive real :(
Axiom 1: P(E) elem N => P(E) >= 0, for all E
Trivially satisfied
Axiom 2: P(Omega) = 1
Satisfied: Omega = N { Inf } elem N
Axiom 3: Sigma additivity. Trivially satisfied since it either includes { Inf } or it doesn't, making the outcome 0 or 1.
Where is the problem ?
I think it's pretty clear that this is the only possible solution, because since N is not closed, there is no way to keep a uniform density other than 0.
This does not seem like it's a very useful solution, but it does seem to satisfy the axioms.