I have a question about Wasserstein space. I am just wondering if the following statement is true
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with $\bar{\Omega}$ being a Polish space and $\bar{\mathbb{P}}$ being atomless measure. Given any $m_0 \in \mathcal{P}_2(\mathbb{R})$ (Borel probability measure with finite second moment) and any $\xi_0 \in L^2(\Omega,\mathcal{F},\mathbb{P}; \mathbb{R})$ with law $m_0$, then for any $m \in \mathcal{P}_2(\mathbb{R})$, there exist $\xi \in L^2(\Omega,\mathcal{F},\mathbb{P}; \mathbb{R})$ such that $$ \mathbb{E}[(\xi_0-\xi)^2] = W_2^2(m_0,m) $$
where $W_2$ denote the Wasserstein distance of order 2.
If this is true, could you provide me a quick proof or reference where I can see the proof and cite this statement?
Thank you!!