You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>41
\sigma-algebras are by definition closed under countable unions, hence the "\sigma".  Otherwise it's just an algebra.  And I'm not positive this is what you're saying, but an outer measure must satisfy

so that if the Vitali sets are in your \sigma-algebra and each have measure 0, then [0,1] has measure 0, which you probably don't want.
>>42
cool, these look interesting and accessible.  Unfortunately I can't read them because I don't have a springerlink subscription :(