Hi everyone,
I have been recently working on metric spaces, doing some exercises about proving properties, integrability... But there's one I have not idea how to prove it (or maybe I should give an counterexample).
I have to prove if there exists a metric space (X,M,$$\mu$$) such that {$$\mu$$(E) : such that E $$\in$$ M} = [0,+$$\infty$$]\Q$$^{+}$$

If anyone has an idea I would be delighted to read u, thanks in advan
Guest

### Re: Proof about metric spaces

It looks like you are simply asking if it is possible to give a metric to the set of all positive irrational numbers. Is that correct?

HallsofIvy

Posts: 341
Joined: Sat Mar 02, 2019 9:45 am
Reputation: 122

### Re: Proof about metric spaces

You can always give the "discrete metric" to any set with d(x, x)= 1, d(x, y)= 0.

HallsofIvy

Posts: 341
Joined: Sat Mar 02, 2019 9:45 am
Reputation: 122

### Re: Proof about metric spaces

HallsofIvy wrote:You can always give the "discrete metric" to any set with d(x, x)= 1, d(x, y)= 0.

I just got that backwards! d(x,x)= 0, d(x, y)= 1 if x is not y.
Guest