# A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric space X) given the 2-Wasserstein metric $W_2$. I was wondering if there is a 'canonical' way of endowing ($\mathcal{P}(X),W_2)$ with a 'nice' probability measure.

More specificaly, let $(X,d)$ be a metric space, and $(\mathcal{P}(X),W_2)$ it's space of probability measures with the 2-Wasserstein metric. Then,

1. Can we set a 'nice' measure, $\mu$, in $(\mathcal{P}(X),W_2)$? (here by nice I guess I mean a non trivial measure that will maybe let us study $(\mathcal{P}(X),W_2,\mu)$ as a metric mesure space. I realize this is vague, a little guidance here would be appreciated)

2. If we can, what conditions on $X$ are required?

3. Is there any other measure that is usually given to $(\mathcal{P}(X),d)$? Where $d$ can be another metric distinct from $W_2$.

Thanks, for the time. Any comments and references are highly appreciated!

-------EDIT--------

(After a while) I posted this question in mathoverflow, where it was answered. Here the link:

https://mathoverflow.net/questions/203499/a-measure-on-the-space-of-probability-measures

• Look up "random measures", their distribution are measures on the measure space under various topologies. – Jorkug Apr 10 '15 at 6:02
• alright, thanks @Jorkug . Do you have any literature recommendation? – Bruce Wayne Apr 10 '15 at 6:36
• Nope, sorry, never read anything comprehensive on this topic nor encountered your particular case. – Jorkug Apr 10 '15 at 7:17
• maybe this question would be better suited for MathOverflow? I'm not sure though – Chill2Macht Apr 2 '17 at 9:53
• @Will2Macht Oh yes! subsequently the question was actually asked and answered there but I forgot to edit here. Thanks for the comment. – Bruce Wayne Apr 3 '17 at 12:33