User:NikelsenH/Measure inversion

In measure theory, a measure inversion is a operation that transforms a measure on the real numbers into another measure

Definition
Let $$ \mu $$ be a measure on the real numbers and let $$ F_\mu $$ be its distribution function.

Then the measure inversion $$ \tilde \mu $$ of $$ \mu $$ is defined as the pushforward measure of the Lebesgue measure $$ \lambda $$ under $$ F_\mu(t) $$:
 * $$ \tilde \mu = \lambda \circ F_\mu^{-1}$$

So for every positive measurable function $$ f $$, the equation
 * $$ \int f(t) \; \tilde \mu(\mathrm dt)= \int f \circ F_\mu(t) \; \lambda(\mathrm dt) $$

holds.