I got a question while reading page 148 of Katok and Hasselblat.
Suppose $\mu$ is an $L_{g_0}$-invaraint Borel probability measure for the translation $L_{g_0}(h) = g_0h$ on a compact metrizable abelian group. Let $\lambda_G$ be the Haar measure. Define for all $g \in G$, $\mu_g := (L_g)_*\mu$ to be the push-forward by $L_g(h):= gh$. Let $E$ be a measurable set with positive Haar measure and finally define $$ \mu_E(A):= \dfrac{1}{\lambda_G(A)}\int_E \mu_g(A)\ d\lambda_G(g). $$
Since $\mathcal{M}(L_{g_0})$, the set of all $L_{g_0}$-invariant Borel probability measures, is convex and $\text{weak}^*$-closed, $\mu_E$ also is both well defined and belongs to the same set. I understand why is the convexity important but why is the $\text{weak}^*$-closed assumption needed?
Update: Is it only needed for the claim of the paragraph before equation (4.2.4) saying that $\overline\varphi_g := \int \varphi \circ L_g$ is continuous in $g$ or the definition/properties of $\mu_E$ also needs it?