Let $X$ be a random variable with $E(X^2) < \infty$, and $Y$ be a binary random variable taking values $\{0,1\}$. The Odds Ratio (OR) between the odds of an outcome at $X=a+\epsilon$ (where $\epsilon$ is a constant) and $X=a$ is
$$\frac{P(Y=0|X=a)P(Y=1|X=a+\epsilon)}{P(Y=0|X=a+\epsilon)P(Y=1|X=a)}$$
where all the probabilities are greater than $0$. A quantity I am interested in is the Average Odds Ratio. However I am having trouble figuring out the right way to define it using the law of the unconscious statistician (LOTUS).
One way to do it is use LOTUS for two jointly distributed random variables.
$$ \int \frac{P(Y=0|X)P(Y=1|Z)}{P(Y=0|Z)P(Y=1|X)} f(X, Z) dXdZ $$
Where $Z=X-\epsilon$. (Notice from the definition of the OR that $X=a+\epsilon$ is the same as $X-\epsilon=a$). I’m not sure about this, though, since the joint distribution $f(X, Z=X-\epsilon)$ is probably ill-defined.
I think that
$$ \int \frac{P(Y=0|X)P(Y=1|X-\epsilon)}{P(Y=0|X-\epsilon)P(Y=1|X)} f(X) dX $$
is possibly better defined, since each of the random variables is a function of $X$.
Are these two definitions equivalent? Is one way better than the other? Any help would be appreciated.