I am trying to prove that the space $\mathcal{P}(\mathbb{R}^n)$ of Borel probability measures over $\mathbb{R}^n$ is separable, convex and most importantly connected.
I can show that $\mathcal{P}(\mathbb{R}^n)$ is a convex and seperable metric space when endowed with the Prokhorov metric (weak convergence, weak star topology). But I get stuck trying to prove that $\mathcal{P}(\mathbb{R}^n)$ is connected.
As $\mathcal{P}(\mathbb{R}^n)$ is convex, it seems intuitive to look at path-connectedness. So it would suffice to prove that for $\mu, \sigma \in \mathcal{P}(\mathbb{R}^n)$ the curve \begin{equation}\gamma: [0,1] \rightarrow \mathcal{P}(\mathbb{R}^n); \lambda \mapsto \lambda\mu + (1-\lambda)\sigma \end{equation} is continuous. But I have problems showing this.
My questions are:
- Is $\mathcal{P}(\mathbb{R}^n)$ connected?
- If not, I only need to this result for probability measures with finite suppport. Does it hold then.
- If $\mathcal{P}(\mathbb{R}^n)$ is connected, does this hold for more general cases? For example complete, separable, connected metric spaces.
I'd be very greatful for some help and sources that deal with this topic.
In any topological vector space the linear operations are continuous, which implies that the curve you wrote is a continuous function of $\lambda$, and ultimately that all convex subsets are connected.
Consider the vector space of all signed measures on $\mathbb R^n$ with the same topology as the one you said in the question. Is it a topological vector space? If it is, then you're done as $\mathcal P(\mathbb R^n)$ is a convex subset of it.