Suppose that we have two independent random variables $V$ and $W$ over $\mathbb{R}$. Suppose that $W$ has a probability density with respect to the Lebesgue measure.

**My question:** Can we find conditions on the density of $W$ the conditional expectation
\begin{align}
\phi_V(t)=E[V|V+W=t],
\end{align}
is unique with respect to a distribution of $V$.

In other words, can we show conditions on $W$ under which $\phi_V(t)=\phi_U(t) , \forall t$ would imply that the distributions of $U$ and $V$ are identical?

This question is very similar in spirit to the uniqueness theorem for the characteristic function.

I feel the following decompositin is the key: \begin{align} \phi_V(t)=E[V|V+W=t]= \frac{E \left[V f_W(t-V)\right]}{E \left[f_W(t-V)\right]}. \end{align}