\begin{align} E[Y] = \sum_{y}yp_Y(y) = \sum_{y}\sum_{x:g(x)=y}p_{X}(x) = \sum_{y} \sum_{x:g(x)=y} yp_{X}(x)\newline = \sum_{y} \sum_{x:g(x)=y}g(x)p_{X}(x) = \sum_{x}g(x)p_{X}(x) \end{align}