Let $\mathcal{X}$ be a sample space, $X$ be random variable that takes values from $\mathcal{X}$, and $x \in \mathcal{X}$ be a data sample.

Can I define a probability distribution $P$ over $\mathcal{X}$ without using the notion of a random variable $X$? That is, define $P_\mathcal{X}$ instesd of $P_X$? Is this a common notation in math, and is it accepted in a research paper in the ML community without further explanation?

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    $\begingroup$ On any non-empty set you can define a sigma algebra and a probability measure. $\endgroup$ – Kavi Rama Murthy Feb 23 at 6:09

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