
Re: the probability theory has holes!
Posted:
Dec 3, 2009 7:55 PM


On Dec 3, 8:53 am, James Burns <burns...@osu.edu> wrote: > Ostap S. B. M. Bender Jr. wrote: > > > On Dec 1, 11:39 am, James Burns <burns...@osu.edu> wrote: > > >>One of the more pleasant surprises on the Internet > >>is finding someone who will admit that they were wrong > >>(when they finally understand that they were wrong). > > >>I wonder, are you going to surprise us? > > > I have a surprise for him: the uniform measure on Z+ > > (or any other infinite countable set) doesn't exist, > > so there can be no valid probability arguments > > of this sort. > > I suppose I understand what you're saying here: > There is no function f: Z+ > R such that both > f(m) = f(n) for all m,n in Z+ > and > Sum_{n in Z+} f(n) = 1 > That's easy enough to prove. Certainly, I am not > disputing that. > > However, it looks to me as though a good argument > very much like the quickanddirty arguments > being used in this thread is possible with > an expanded definition of a measure on Z+, > a definition much like that for generalized > functions, with the Dirac delta function as a > particular example. > > As I understand it, the Dirac delta is described > as a function that is 0 everywhere except one > point, but, when it is integrated across that > point, we get 1. This is obviously (and provably) > impossible, as it is stated. >
Well, I suppose you can define some relevant probability measures in the realm of functional analysis, where the the Dirac deltas live.
> > The workaround that > I am familiar with uses a sequence of > functions < d_n > such that > d_n(x) > 0, as n > infinity, for x <> 0, > and > INT{inf, +inf} d_n(x) dx = 1, all n > There are a lot of choices for such a sequence. > > A rigorous argument involving Dirac deltas > would solve some problem using a convenient > choice for the d_n, get an answer involving > n, and then take the limit as n > infinity > as the "true" answer. The operations behind > what we call the Dirac delta are completely > unexceptional, but they have the same effect > as the provably impossible function. > > My question is: why can't something similar > be done to create a uniform measure on Z+, > or, at least, something that would give us > the answers a uniform measure would, if such > a thing existed? > > For concreteness, suppose > f_n : { m in Z+  m =< n } > R > where > f_n(m) = 1/n > > Suppose we ask what the probability that a > "random number" is not divisible by 2, 3, or 5 > (for example), and get a constant plus some > bounds that > 0 as n > infinity. Without > doing very much in the way of calculation, I > feel confident that the constant would be > (1/2)(2/3)(4/5). > > What is wrong with this argument? >
Probably not much.
Howevewr:
1. I was replying to the premise of this thread:
"the probability theory has holes!"
2. How is this going to help one with the twin primes conjecture?

