Before revision:  In these settings, the probability measure of interest is induced by an unknown map from a known probability space to the real line, i. e., the measure of interest is a pushforward of another known measure. In computation, we do not know the true map , but only an approximate one. In the field of UQ, the{\em   generalized Polynomial Chaos (gPC) method is widely popular and yields excellent approximations of the map and its moment .
After revision:  In these settings, the probability measure of interest is induced by an unknown map from a known probability space to the real line, i. e., the measure of interest is a fixed function of the parameters, and can therefore be considered as a random variable as a well. Often, the quantity of interest map is not explicitly known, and so the computational problem is to find its ``right'' approximation (surrogate model). For the goal of approximating the{\em generalized Polynomial Chaos (gPC) method is widely popular and yields excellent approximations of the map and its moment .