Is there a closed-form expression (or simpler expression) for the survival function of the distribution itself: $$V(z)=\int\mathbb{1}_{\{\mathbf{x}\mid p(\mathbf{x})>z\}}p(\mathbf{x})\mathrm{d}\mathbf{x}$$ where $z\in \mathbb{R}$, $\mathbf{x}\in \mathbb{R}^n$ and $p(\mathbf{x})$ is a probability distribution on $\mathbb{R}^n$.
The question is motivated by the fact that for $x\in \mathbb{R}$ $$\int F(x)\mathrm{d}F(x)=\frac{1}{2}$$ where $F(x)$ is a CDF.