Let $(X,{\goth X})$ be a measurable space. Let $\tau $ be a bi-measurable bijection from $X$ onto $X$. Let $\varphi $ be a measurable application from $X$ to a second countable locally compact group $G$. We denote by $\tau _{\varphi }$ the extension of $\tau $, induced by $\varphi $, to the product space $X\times G$. We describe the positive $\tau _{\varphi }$-invariant and ergodic measures on $X\times G$. We also obtain a generalization of the cocycle reduction theorem of O. Sarig [?] to a general second countable locally group.