Is it possible to put a probability space into the Hierarchy of mathematical spaces?

Question

This is the figure of Hierarchy of mathematical spaces in this wiki page.

a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers

so, is it possible to put a probability space into this Hierarchy of mathematical spaces?

Answer 1

It doesn‘t fit into this total order: a probability space is simply a measure space such that the measure of the whole space is $1$. A measure can be put on any set and a priori it doesn‘t have any relation to topological spaces or metric spaces.

Answer 2

Here's what it looks like:

Probability spaces are a special kind of measure spaces (just those with total measure 1). There's no intrinsic relation between measures and topologies, or between measures and metrics. You can put a measure (even a probability measure) on any set you want (Exception: you can't put a probability measure on the empty set).