Perturbative approach starting from a probability distribution approximated form

Question

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 ?

Answer 1

$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Translated into a standard language of probability theory, your question appears to be as follows:

Let $p$ and $q$ be two probability density functions (say on $\R$) such that $$|p-q|\le\ep\tag{0}$$ for some real $\ep>0$. Let $Y$ be a Lebesgue-measurable function from $\R$ to $\R$. Does it then follow that $$\Big|\int pY-\int qY\Big|\le C\ep\tag{1}$$ for some real $C\ge0$?

The answer is this: inequality (1) holds for some real $C>0$ and all $p$ and $q$ satisfying condition (0) if and only if $$C_Y:=\int|Y|<\infty;$$ moreover, the best possible constant factor $C$ in (1) is $C_Y$.

Indeed, on the one hand, if (0) holds, then $$\Big|\int pY-\int qY\Big|=\Big|\int(p-q)Y\Big|\le\int|p-q|\,|Y|\le\int\ep\,|Y|=C_Y\ep,$$ so that (1) holds with $C=C_Y$.

On the other hand, suppose that $$p=\ep\,1_{[0,1/\ep]},\quad q=\ep\,1_{[-1/\ep,0]},$$ $Y\ge0$ on the interval $[0,1/\ep]$ and $Y=0$ outside this interval. Then (0) holds, and $$\Big|\int pY-\int qY\Big|=\int pY=\ep\int Y=C_Y\ep.$$

So, $C_Y$ is the best possible constant factor $C$ in inequality (1) given condition (0).