Let $G$ be a compact group, and $H$ be Hilbert. Let $U$ be a unitary representation of $G$ on $H$(it's a group homomorphism from $G$ into the unitary operators of $H$). Define $E$ on $\mathbf{B}_0(\mathcal{H})$(compact operators on $H$) by $$E(T) := \int_G U_xTU_x^* dx$$Because $\mathcal{H}$ is Banach, then $\mathbf{B}(\mathcal{H})$ is Banach so we can use the Bochner integral. I am not used to the Bochner integral, can someone help me integrate this? I am trying to show that $E(T)$ is a compact operator. I have already established that $U_xTU_x^*$ is compact due to $T$ being compact and unitary representation.
I would like to mention that $x \mapsto U_xTU_x^*$ is norm continuous on the norm of $\mathbf{B}(H)$(edit)
Another edit, we are using thee Haar measure on $G$