Let \(\mathcal{M}_{\mathbb{P}}(X)=\{\mathbb{Q} \ll \mathbb{P}: X\) is a \(\mathbb{Q}\)-martingale \(\}\). For any convex set \(\mathcal{K}\), we denote by \(\operatorname{ext}(\mathcal{K})\) the set of extremal points of \(\mathcal{K}\). Prove that

\[\operatorname{ext}\left(\mathcal{M}_{\mathbb{P}}(X)\right)=\operatorname{ext}(\mathcal{M}(X)) \cap \mathcal{M}_{\mathbb{P}}(X)\]

An open question is: does the equality

\[\operatorname{ext}\left(\mathcal{M}_{\mathbb{P}}^{e q}(X)\right)=\operatorname{ext} \mathcal{M}(X) \cap \mathcal{M}_{\mathbb{P}}^{e q}(X)\]

where \(\mathcal{M}_{\mathbb{P}}^{e q}(X)=\{\mathbb{Q} \sim \mathbb{P}: X\) is a \(\mathbb{Q}\)-martingale \(\}\), hold?