Definition

With the same convention defined in the Wasserstein metric the Wasserstein space of order $p$ is defined as $$P_{p}(\mathcal{X}):=\left\{ \mu \in\mathcal{P}(\mathcal{X})\middle| \int\limits _{\mathcal{X}}d(x_{0},x)^{p} \, \mu(dx)<+\infty \right \}\quad x_{0}\in\mathcal{X}$$where $x_{0}\in\mathcal{X}$ is arbitrary. This space does not depend on the choice of the point $x_{0}$. Then $W_{p}$ defines a finite Metric on $P_{p}(\mathcal{X})$.

The Wasserstein space is the space of Probability Measures which have a finite moment of order $p$.