Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process
$$X_{t}(\mathbf{x})=\mathbf{x}(t)$$
and the canonical filtration $\mathcal{F}_t=\sigma(X_s, 0\le s\le t)$. Now suppose there exists some probability $\mathbb{P}$ on $\mathbf{D}$ s.t. $X$ is a martingale. Let $V$ denote some supermartingale under $\mathbb{P}$ (without any assumption about the regularity of its pathes), now we define
$$V^+_t=\limsup_{s>t, s\in\mathbb{Q}}V_s$$
Then $V^+$ is again a supermartingale w.r.t to the filtration $\{\mathcal{F}_t^+\}_{0\le t\le 1}$. Since by a modification of $V^+$ (since $E^{\mathbb{P}}[V^+_t]$ is right-continuous), we can get a cadlag supermartingale $\tilde{V}$, my question is whether we have
$$V_0=esssup_{\mathbb{P}}(\tilde{V}_0)=esssup_{\mathbb{P}}(V^+_0)?$$
Thanks a lot for the reply!
