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I see nothing wrong with Krishna's expression. If I had to take a guess, it would be that either Menezes and Monteiro define their $f_{Y|X}(y|x)$ differently or they simply forgot to adjust the density on $Y_1 &lt; x$. Both expressions are supposed to be the same.

If you have some random variable $Y$ with cdf $F$, density $f$ and support $[a,b]$
, the following two notations say the same thing
$$\int v(y) f(y) dy  = \int v(y) d F(y) \quad  \mbox{for any function } v(y).$$
If you condition $Y$ on being $Y&lt;x$, you have to adjust the cdf because the new support only goes up to $x$,
$$ F_{Y|Y&lt;x} (y) = \begin{cases}
0 \quad &amp;\mbox{if } y &lt;a, \\
\frac{F(y)}{F(x)} \quad &amp;\mbox{if } y \in [a,x],\\
1 \quad &amp;\mbox{if } y &gt;x.
\end{cases}$$
That way you have an expression that is one for $y=x$.

Hence,
$$E[v(Y_1,Y_1)|X_1=x,Y_1&lt;x]= \int_0^x v(y,y)d\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)}
= \int_0^x v(y,y)  \cfrac{ g_{Y|X}(y|x)}{G_{Y|X}(x|x)}   d y.  $$