Measure Space without any form of metric

Question

I'm with a difficult question that I was thinking . Any help will be useful

We know that the Urysohn's Theorem states that: every Hausdorff second-countable regular space is metrizable.

There are other metrization theorems, but this is the most famous and easy example that I remember.

I'm trying to make a Measure Space in a non-metrizable space. I think in use Urysohn's theorem, but the problem with this theorem is that the converse is not true. There are examples of non-second countable topological spaces that are metrizable.

So I don't know how to proceed, there are others metrizable theorems and colloraries of the Urysohn, but I can't tink in any form that how can I make progress with this theorems

Answer 1

Trivial remark:

If we have a finite or countable measure space $X$ with $\sigma$-algebra $\mathcal{M}$, then this can only be the Borel-algebra of a metrisable topology on $X$ iff $\mathcal{M}=\mathscr{P}(X)$.