Recently I tried to put a measure on the powerset of the natural numbers , but it was pointed out in a comment that the map I defined is not even finitely additive.

**Question**: Is there a map with the following properties:

- ;
- if such that then ;
- is translation-invariant, that is for and where .

The solution that occurs to me relies on the concept of Banach limit

http://planetmath.org/banachlimit

So choose any Banach limit $\Phi$, i.e., a bounded linear functional in the space of bounded sequences such that it is invariant under shifts. Then one just substitute the limit operator in the previous entry of the blog with the operator $\Phi$. You get immediately the three properties for $\mu$.

see from your recent cstheory.se post you are interested in P vs NP. lots of musings on that on my blog & also consider dropping by this chat room for further discussion

Yes, and you can even enforce $\mu$ to be $(-1)$-homogeneous: see http://mathoverflow.net/questions/209706/ or Remark 3 in http://arxiv.org/abs/1506.04664.