Let $(\Xi,\mathcal{E})$ be a measurable space, we denote $\mathcal{M}(\Xi)$ the set of all probability measures supported on $\Xi$. In an analogous way $\mathcal{M}(\Xi^{N})$ is the set of all probability measures supported on $\Xi^{N}$ where $(\Xi^{N},\mathcal{E}^{N})$ is measurable space with $\mathcal{E}^{N}$ as the product $\sigma$-algebra.
Question: If we denote $\mathcal{X}=\mbox{span}\left(\mathcal{M}(\Xi^{N})\right)$, is it correct to say that $$ \mathcal{X}=\mbox{span}\left(\mathcal{M}(\Xi^{N})\right)=\mbox{span}\left(\mathcal{M}(\Xi)^{N}\right)=\mbox{span}\left(\otimes_{i=1}^{N}\mathcal{M}(\Xi)\right)?$$ Furthermore, if we denote $\mbox{cone}(S)$ as the cone generated by the set $S$, then is it correct to say that $$\mathcal{C}:=\mbox{cone}\left(\mathcal{M}(\Xi^{N}) \right)=\mbox{cone}\left(\mathcal{M}(\Xi)^{N} \right)=\mbox{cone}\left(\otimes_{i=1}^{N}\mathcal{M}(\Xi) \right)?$$