I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic observable $Y = Y(x)$; $Y$ is a generic function of the random variable $x$, and so it is also a random variable. If I want $E(y) \equiv \int P_x(x') Y(x') dx' $ I expect an expression as:
$E(y) = E^{app}(y) + O(\epsilon)$, where $E^{app}(y) \equiv \int P_x^{app}(x') Y(x') dx' $.
How can one shows this from a formal point of view ?