Yes, this is called 'majorization' or '2-stochastic domination' of measures (first term is used in analysis, second in probability). The idea is very simple: we partition the measure $\mu$ on several summands $\mu=\sum \mu_i$ and replace each $\mu_i$ to its mean value.

Yes, this is called 'majorization' or 'second order stochastic dominance' of measures (first term is used in analysis, second in probability). The idea is very simple: we partition the measure $\mu$ on several summands $\mu=\sum \mu_i$ and replace each $\mu_i$ to its mean value.