Can someone possibly explain how to >>make sense<< of the following identity:
$\int \frac{\partial \ max_d \{ u(x,d) + \epsilon(d) \} }{\partial u(x,d)} q(d\epsilon \lvert x) = \int I\{d = \arg \max_h [u(x,h) + \epsilon(h)] \} q(d \epsilon \lvert x)$
I completely fail to see how taking the partial derivative under the first integral results in the expression with the indicator function?
The expression simply says that when you change a function, u(x,d), you only change the max of that function if you change u(x,d*) where d* is the argmax of u(x,d) with respect to d.