In probability and statistics literature it seems that people often describe random variables in terms of their distributions and omit any discussion of sample spaces and the underlying probability triple that the random variable is defined on.
For instance, it is common to describe a random variable that has the Dirichlet distribution as,
$$ X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha ) $$
i.e., a random variable whose values live on a $K-1$ simplex and whose pdf is
$${\displaystyle f\left(x_{1},\ldots ,x_{K};\alpha _{1},\ldots ,\alpha _{K}\right)={\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}}$$
The domain of $X$ is completely omitted in such a description. This seems to blur the line between random variables and probability distributions.
My question isn't super focused. I am more just confused by the blurring of these two concepts. I have no idea what the sample space would even be for a r.v. with a Dirichlet distribution. I guess I could always just say that X is the identity function and that $\Omega$ contains the points on the $K-1$ simplex.
Are sample spaces unimportant once we have the distribution of a random variable? How should we think about these things?
