If the two random variables $X$ and $Y$ are independent, I need to determine the two sets A and B that will make the following equation true
$Pr ((X^{-1}, Y^{-1}) (A \times B)) = \int _{-\infty} ^{t} f_{Y}(y) F_{X} (y) dy $
where $f_{Y}$ is the PDF of $Y$, and $F_{X}$ is the CDF of $X$
Can we write the event that corresponds to this integration in this form using the preimage of the random variables and the product of two sets?
Since
$$F_{X} (y)=\int_{-\infty}^y f_X(x)dx$$ we have
$$\int _{-\infty} ^{t} f_{Y}(y) F_{X} (y) dy=\int _{-\infty} ^{t} f_{Y}(y) \left[\int_{-\infty}^y f_X(x)dx\right]dy=$$$$=\int_{-\infty}^t\int_{-\infty}^yf_Y(y)f_X(x)\ dxdy=\iint_{\{(x,y):\ x\leq y,\ y\leq t\}}f_{X,Y}(x,y)\ dx dy.$$ This is a double integral of the joint probability density function over the region showed below:
That is
$$\int _{-\infty} ^{t} f_{Y}(y) F_{X} (y) dy=P(Y\leq t\ \cap\ X\leq Y).$$
So the given integral cannot be written in the form of
$$P((X^{-1}, Y^{-1}) (A \times B)). $$