Let $Y$ be a compact abelian group and let $m$ be the haar measure. Suppose that $f:X \to Y$ is measurable and let $T:X \to X$ be an invertible measure preserving transformation. Show that the entropy of $\tau(x,y)=(T(x),y+f(x))$ is equal to $h(T)$.
I know that for skew products $(T(x),S_xy)$ the entropy is equal to $h(T)+h_T(S)$ where $h_T(S)$ is the fiber entropy. Here $S_xy=y+f(x)$. My initial thought is to estimate the sizes of the partitions $\beta_1^n(x)=S_{x}^{-1} \beta \vee S_{x}^{-1}S_{Tx}^{-1} \vee \dots \vee S_{x}^{-1} \dots S_{T^{n-1}x}^{-1} \beta$ and show that it must be bounded by $Kn$. Then $h_T(\beta,S)=\lim_{n \to \infty} \frac{1}{n}\int_X H( \beta_0^n(x)) \to 0$.