Given two random variables, both continuous, $X$ and $Y$, we know that $P(X=x)=P(Y=y)=0$. Is it an abuse of notation or is it correct to have the following?
$$P(X<u | Y=y) > 0$$
In particular, can we claim the following?
$$P(X<u | Y=y) = \frac{P(X<u, Y=y)}{P(Y=y)}$$
We have \begin{align} F_X(u) = P(X< u) &= \int_{y=-\infty}^{\infty} \quad \int_{x=-\infty}^u f(x,y) dx \quad dy \\ &= \int_{y=-\infty}^{\infty} \quad \frac{ P( X < u, y < Y < y+dy)}{dy} \quad dy \\ &= \int_{y=-\infty}^{\infty} \quad { P( X < u| y < Y < y+dy)} f_Y(y)dy{} \\ \end{align}
In the expression above, it looks like we can claim the following
$$P(X<u | Y=y) = P(X<u | y < Y < y +dy)$$
however, in the left hand side we do not have $dy$ and in the right hand side we do?
What about the following?
$$P(X<u | Y=y) = \frac{P(X<u, y < Y < y+dy)}{P(y < Y < y+dy)} =\frac{\int_{x=-\infty}^u f_{X,Y}(x,y)dxdy}{f_Y(y) dy} $$
Is the best to say
$$P(X<u | Y=y) = \lim_{\Delta y \rightarrow 0} \frac{P(X<u, y < Y < y+ \Delta y)}{P(y < Y < y+ \Delta y)} $$
or
$$P(X<u | Y=y) = \lim_{\Delta y \rightarrow dy} \frac{P(X<u, y < Y < y+ \Delta y)}{P(y < Y < y+ \Delta y)} $$
?
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