How to find \begin{align} \limsup_{a \to 0} E[V | aV+W] \end{align} where $V$ and $W$ are independent. We can assume that $E[V^2], E[W^2] <\infty$.
I think the following must \begin{align} \limsup_{a \to 0} E[V | aV+W]=E[V] \text{ a.s.} \end{align}
I found this property called martingale convergence that says: If $E[V] <\infty$ then \begin{align} \lim E[ V | \mathcal{H}_n] \to E[ V | \mathcal{H}] \end{align} if $ \mathcal{H}_n$ is a decreasing sequence (nested) of sub-sigma algebrals and $ \mathcal{H}=\cap_n \mathcal{H}_n$,
But how to apply this? Or this is ever a correct tool to use?
{\bf Edit 1:} See counter example A.S. Now suppose we assume that $V$ is Gaussian random variable.