  In many applications including financial risk measurement, copulas have shown
to be a powerful building block to reflect multivariate dependence between
several random variables including the mapping of tail dependencies.
  A famous key result in this field is Sklar's Theorem. Meanwhile, there exist
several approaches to prove Sklar's Theorem in its full generality. An elegant
probabilistic proof was provided by L. R\"{u}schendorf. To this end he
implemented a certain "distributional transform" which naturally transforms an
arbitrary distribution function $F$ to a flexible parameter-dependent function
which exhibits exactly the same jump size as $F$.
  By using some real analysis and measure theory only (without involving the
use of a given probability measure) we expand into the underlying rich
structure of the distributional transform. Based on derived results from this
analysis (such as Proposition 2.5 and Theorem 2.12) including a strong and
frequent use of the right quantile function, we revisit R\"{u}schendorf's proof
of Sklar's theorem and provide some supplementing observations including a
further characterisation of distribution functions (Remark 2.3) and a strict
mathematical description of their "flat pieces" (Corollary 2.8 and Remark 2.9).
