Sorry for being so loose and informal about the mathematical objects that I am using. If your answer depends on some "regularity" conditions you can state them, that would be very useful.
I am working with a continuous random variable $w$, which is distributed over the support $[\underline{w},\overline{w}]$ according to the CDF $G(w)$. I have infinite observations of $w$, say a "continuum of mass one" and of a "well-behaved" function $t(w)$. I want to "aggregate" this observations, I am doing that in the following way:
$$T(w)=\int_{\underline{w}}^{\overline{w}}t(w)dG(w)$$
Would this way of aggregating be correct?
Also, let's say I want to compute $\frac{\partial T(w)}{\partial t(w)}$. Would the correct to do this would be to just take the derivative under the integral sign and obtain:
$$\frac{\partial T(w)}{\partial t(w)}=\int_{\underline{w}}^{\overline{w}}1\cdot dG(w)=G(\overline{w})-G(\underline{w})=1-0=1$$
since I have a continuum I somehow expect to obtain marginal impacts of zero, but this is of course not the case.