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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?

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We use these different concepts because they are natural and appropriate for different sorts of problems and results. The general notion of probability space is the natural context for basic notions of probability theory, like countable additivity, conditional probabilities, independence, and random variables. Cumulative distribution functions are a convenient way to describe probability spaces in which the underlying set is $\mathbb R$ (or $\mathbb R^n$) and the $\sigma$ algebra of measurable sets is the Borel algebra or a completion of it. Such spaces arise naturally, especially when one takes the image of a general probability measure under a measurable function to $\mathbb R$ (in which case one calls the function a random variable and one calls the cdf of the pushed-forward measure the cdf of that random variable). Cumulative distribution functions summarize such measures quite efficiently, taking advantage of the fact that knowing the measure of sets of the form $(-\infty,a]$ (or products of such sets in $\mathbb R^n$) determines the measures of all Borel sets. Probability density functions work for an even more specialized context, but in that context they serve to make many computations easier.

Your suggestion concerning "some notion regarding symmetry" is probably intended for the context of $\mathbb R$ or $\mathbb R^n$ or some similar space on which a (specified) symmetry group acts. I don't see any meaningful notion of symmetry for completely general probability distributions. You're probably thinking of a context where cumulative distribution functions make sense. But for many (probably most) forms of symmetry, if probability density functions are available then they will show the symmetry more clearly. (Consider, for example, rotational symmetry in $\mathbb R^2$.

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