probably boring set theory questions

[anybody_mcc]

I'm stuck with some set theory problems , so anyone who is interested or just knows the answer is welcome :

First of all some definitions ( i'm not sure if i am using right english terms ) :
a)By measure i will mean finite-additive measure ( not σ-additive measure ).
b)Density of a set A ( subset of ω ) = lim A(n)/(n+1) if such exists , where A(n) = |{ 1,2,3, .... n } ∩ A|
c)Frechet ( finite ) ideal : ideal of all finite subsets of ω.

We know that every measure on P(ω) determines some ideal on ω :
if we have measure μ , then { A ; μA = 0 } is an ideal on ω.

However Frechet ideal and ideal of sets of density 0 are not determined by any measure. Is that really true ? How can it be proven ?

[anybody_mcc]

i hope the special signs i used are readable ?

[Kuroneko]

First of all some definitions ( i'm not sure if i am using right english terms ) :
a)By measure i will mean finite-additive measure ( not σ-additive measure ).
b)Density of a set A ( subset of ω ) = lim A(n)/(n+1) if such exists , where A(n) = |{ 1,2,3, .... n } ∩ A|

Not n, but n+1? Well, it shouldn't actually matter.

c)Frechet ( finite ) ideal : ideal of all finite subsets of ω.

Alright.

We know that every measure on P(ω) determines some ideal on ω : if we have measure μ , then { A ; μA = 0 } is an ideal on ω.

Let's take this as a definition of the ideal of a measure.

However Frechet ideal and ideal of sets of density 0 are not determined by any measure. Is that really true ? How can it be proven ?

What's wrong with μ(A) = 0 for every finite set A and μ(A) = \infinity otherwise? That's a finite-additive measure whose ideal is the Frechet ideal. A similar trick should produce a measure whose ideal is the set of zero-density sets. Or do you have some sort of requirement that μ(ω) be finite or anything of that sort?

[anybody_mcc]

However Frechet ideal and ideal of sets of density 0 are not determined by any measure. Is that really true ? How can it be proven ?

What's wrong with μ(A) = 0 for every finite set A and μ(A) = \infinity otherwise? That's a finite-additive measure whose ideal is the Frechet ideal. A similar trick should produce a measure whose ideal is the set of zero-density sets. Or do you have some sort of requirement that μ(ω) be finite or anything of that sort?

Sorry , i really forgot to define the measure as a function from P(ω) to R.

[Kuroneko]

Even under that nonstandard definition, something is missing. There isn't any problem with μ(A) = 0 for finite A and μ(A') = 1 for its complement. Can you state your full criteria for what constitutes an acceptable measure?

[anybody_mcc]

ok , measure :
function η from P(ω) to R is measure if
1) η0 = 0 , ηω > 0
2) A is subset of B then ηA <= ηB
3) A ∩ B = 0 then η(A U B) = ηA + ηB

the problem with your example ηA = 0 for finite and ηA' = 1 for complement of A is , i think , in the requirement that η has to be defined on the whole P(ω) , so there is a problem , what to do with odd numbers for example. This could probably be resolved with fractions , but then you have problem with non-finite sets that have 0 density , for example primes. What would be their measure ?

[Kuroneko]

the problem with your example ηA = 0 for finite and ηA' = 1 for complement of A is , i think , in the requirement that η has to be defined on the whole P(ω) , so there is a problem , what to do with odd numbers for example.

There is no such requirement in measure theory--for example, the fact that there are lots of non-measurable sets in R^3 in the partial cause of the famous Banach-Tarski paradox. If you want to add this as a requirement, I'm not (yet) certain whether there would be a measure or not.

[anybody_mcc]

the problem with your example ηA = 0 for finite and ηA' = 1 for complement of A is , i think , in the requirement that η has to be defined on the whole P(ω) , so there is a problem , what to do with odd numbers for example.

There is no such requirement in measure theory--for example, the fact that there are lots of non-measurable sets in R^3 in the partial cause of the famous Banach-Tarski paradox. If you want to add this as a requirement, I'm not (yet) certain whether there would be a measure or not.

true , but now i'm thinking about it , it is not really independent requirement , it follows from 1) and 2) in the definition , i think , since emty set is subset of every non-emty A , so η0 <= ηA , and that should imply that there is ηA defined. I think that with this definition there really is no measure for Frechet ideal , but i don't know how to prove this.