We want to compute the following quantity \begin{align} \int E[V|W=u] f_U(u) du \end{align} For random variables $ V,W,U$ where $V$ and $U$ are independent and $U$ is absolutely continuous r.v.
Is the following chain of equalities correct \begin{align} \int E[V|W=u] f_U(u) du = E_U[ E_{V|W}[V|W] ]=E_V[V] \end{align}
Also, this problem well defined?