I'm trying to minimise this function. $$R(h)=\int_{(x,y)\in(\chi,Y)}{L(h(x),y)dP_0(x,y)}$$ where L(z,y) is just $(z-y)^2$
I tried to differentiate wrt h and set the derivative to zero, but I'm getting confused on how to successively put the $dh$ inside the integral. This is my attempt so far. $$\frac{dR(h)}{dh}=\int2(h(x)-y)dP_0(x,y)=0$$ $$\int{h(x)dP_0(x,y)}=\int{ydP_0(x,y)}$$
The h(x) I get doesn't match the result, I'm supposed to get, which is $$h(x)=\int{ydP_0(x,y)}=E[Y|X=x]$$