Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.
Is it true that $E[X|Y < y]$ equals:
$$ \displaystyle \int_{-\infty}^\infty \int_{-\infty}^y x \frac{f_{XY}(x,y)} {f_{Y}(y)} dydx $$
since $f_{XY}(x | y) = \frac{f_{XY}(x,y)} {f_{Y}(y)} $.
$$E[X\mid Y\lt y]=\frac1{F_Y(y)}\,\int_{-\infty}^{+\infty}x\int_{-\infty}^yf_{X,Y}(x,t)\,\mathrm dt\,\mathrm dx.$$