Let $X,Y_1,\cdots, Y_n$ be $n+1$ random variables with joint density function $f(x,y_1,\cdots, y_n)$. Assume that $Y_1,\cdots, Y_n$ are conditionally i.i.d given $X$. That is there exists a conditional density function $h(y|x)$ such that $$f(y_1,\cdots,y_n|x)=\prod_{i=1}^n h(y_i|x).$$ Assume $$y\mapsto\frac{h(y|\overline{x})}{h(y|\underline{x})}$$ is increasing for all $\overline{x}>\underline{x}$.
Now, we know that for each $i$, $E(X|Y_i)=g(Y_i)$ where $$g(y)=\int x\frac{h(y|x)f_X(x)}{\int h(y|\tilde{x})f_X(\tilde{x})\mathrm{d}\tilde{x}}\mathrm{d}x$$ and $f_X$ is the marginal distribution of $X$. Moreover, given the conditions on $h$, we know $g$ is increasing. Therefore $$\max\{E(X|Y_1),\cdots,E(X|Y_n)\}=g(\max\{Y_1,\cdots,Y_n\}).$$
My question is whether $g(\max\{Y_1,\cdots,Y_n\})$ is a version of $E(X|\max\{Y_1,\cdots, Y_n\}).$ Measurability condition is of course satisfied, but I don't know whether the other condition in the definition of conditional expectation is also satisfied or not.