# Can I define a probaility distribution without defining a random variable?

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?

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