In this work we introduce a novel approach of construction of multivariate cumulative distribution functions, based on cyclical-monotone mapping of an original measure $μ\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ to some target measure $ν\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ , supported on a convex compact subset of $\mathbb{R}^d$. This map is referred to as $ν$-Brenier distribution function ($ν$-BDF), whose counterpart under the one-dimensional setting $d = 1$ is an ordinary CDF, with $ν$ selected as $\mathcal{U}[0, 1]$, a uniform distribution on $[0, 1]$. Following one-dimensional frame-work, a multivariate analogue of Glivenko-Cantelli theorem is provided. A practical applicability of the theory is then illustrated by the development of a non-parametric pivotal two-sample test, that is rested on $2$-Wasserstein distance.