Let $(C,G,\alpha)$ be a $C^*$ dynamical system where $G$ is a locally compact Hausdorff topological group. and let $C_c(G,A)$ be the collection of compactly supported continuous functions from $G \to A$.
Let $f,g \in C_c(G,A)$. Then I want to prove that two integrals are equal. LHS part is $\int \Delta(t^{-1})\alpha_{ts}([f((ts)^{-1}]^*)\alpha_t(g(t^{-1}))dt$.
RHS is $\int \Delta(t^{-1})\alpha_t(f(t^{-1})^*) \alpha_{ts^{-1}}(g((ts^{-1})^{-1}))dt$.
I tried to make the transformation $t \to ts$. But then there is a $\Delta(s^{-1})^2$ factor comes. I dnt know where I am making the mistake.