Conditional expectation $\mathbb E[f(U,V)\mid\sigma(V)]$ when $f$ is only measurable in the first coordinate and $U$, $V$ are independent.

Question

Suppose $U$, $V$ are independent random variables, $f$ is only measurable in the first coordinate and if we provide the condition that $f(U,V)$ is measurable in the probability space, can we show that $\mathbb E[f(U,V)\mid\sigma(V)]=g(V)$, where $g(v):=\mathbb E[f(U,v)]$?