Certain properties of triangular transformations of measures

We study the convergence of triangular mappings on ${\mathbb R}^n,$ i.e., mappings $T$ such that the $i$th coordinate function $T_i$ depends only on the variables $x_1,\ldots,x_i.$ Weshow that, under broad assumptions, the inverse mapping to a canonical triangular transformation is canonical triangular as well. An example is constructed showing that the convergence in variation of measures is not sufficient for the convergence almost everywhere of the associated canonical triangular transformations. Finally, we show that the weak convergence of absolutely continuous convex measures to an absolutely continuous measure yields the convergence in variation. As a corollary, this implies the convergence in measure of the associated canonical triangular transformations.