Is the space of probability distributions an infinite dimensional space?

Question

Is the space of probability distributions an infinite dimensional space? If so, would you explain how?

This is a follow up question to an answer to this question.

Answer 1

Already the probability distributions on $\mathbb N$ are an infinite-dimensional convex set since the Dirac measures at $1$, $2$, ..., $n$ are linearly independent, for every $n$.

Answer 2

The space of probability distributions is not a linear space, so you presumably do not mean linear dimension.

The space $\mathcal P([0,1])$ of probability measures on $[0,1]$, for example, has infinite Hausdorff dimension. Perhaps that is what you mean. We use a standard metric on the space $\mathcal P$ for this. For example: $$ D(\mu,\nu) = \sup \left|\int f \,d\mu - \int f\,d\nu\right| $$ where the sup is over all functions $f \colon [0,1] \to \mathbb R$ bounded by $1$ and Lipschitz constant $\le 1$.