Hey guys, I have a pretty basic question that I want to be sure of. I'm taking a probability over an input selected uniformly at random from binary strings of length $l(n)$. I would like to compare the conditional probability of a function taking a value given that its input is in a subset of this set to the probability of the function taking this value over input selected uniformly at random from the subset.

That is, I want to see if the given equality is valid: $\Pr_{w \leftarrow U_{l(n)}} \left[g\left(A(w)\right) = w \mid w \in g(U_n)\right] = Pr_{w \leftarrow_R g(U_n)} \left[g\left(A(w)\right) = w \mid w \in g(U_n)\right]$

Is this true?