is used to describe "mass transportation"—moving probability mass from one distribution to another to measure risk severity and propensity. instead of just is simpler for introductory statistics,
E[h(X)]=∫Rh(x)PX(dx)cap E open bracket h of open paren cap X close paren close bracket equals integral over the real numbers of h of x cap P sub cap X open paren d x close paren
is governed by the . If the distribution of
, we are referring to the of this measure. This notation is particularly common in measure-theoretic probability and is used to represent integration over the space of Connection to Probability Densities The relationship between the measure and a standard probability density function
: Advanced papers on neural network estimators use measure-theoretic notation to prove convergence rates and universal consistency of models. Optimal Transport : In fields like risk quantization ,