I am noticing some probability theory notions like the space of Borel probability measures on some specific metric spaces. But then I cannot quite understand the reason of defining different kinds of "spaces". Concretely, we might have the following spaces in consideration:
(1) the space of probability measures;
(2) the space of cumulative distribution functions;
(3) the space of probability distribution functions;
I am not sure if some certain property like convergence or continuity of one space would imply some property of the other space. Nevertheless, except the reason that not all probability measures are defined on the real line, what could be the intuitive reasons to define different notions of spaces.
One example I am considering is the following. Suppose we want to define some notion regarding "symmetry". Then, we could start from defining what would be a symmetric probability measure. Then, this symmetry property would be inherent to the corresponding cumulative probability function, if the probability measure is a Stieltjes measure. Also, it is also equivalent to define symmetry by using the corresponding probability density function. Is there any counter-example that shows there is no simple way to induce the symmetry property from one element in one space to one element in another space?