This is a very basic question, and maybe a silly one, but I'm struggling to understand the actual definition of a probability distribution.
In Wasserman's "All of Statistics", for example, he says that "if $\Omega$ is finite and if each outcome is equally likely, then $\mathbb{P}(A) = |A| / |\Omega|$, which is called the uniform probability distribution."
This suggest to me that a probability distribution refers specifically to the probability measure chosen, i.e. a probability distribution is the measure associated with a probability space. Yet on Wikipedia and elsewhere online, a probability distribution is only defined within the context of some random variable, e.g. "A probability distribution is a function that describes the possible values of a random variable and their associated probabilities". So my question is, is a probability distribution the measure on a given space, or is it the pdf/pmf of a given random variable? When we say we "draw a random variable" from a probability distribution, does that mean we construct a random variable from a given probability space with that probability measure, or that our random variable has that pdf/pmf? Is this a meaningful distinction?