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Certain properties of triangular transformations of measures. (English) Zbl 1199.28047
Summary: 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 \(x_1,\dots,x_i\). We show 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.
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G12 Measures and integration on abstract linear spaces
60B11 Probability theory on linear topological spaces