$L_1$ mean ergodic theorem for the action of compact group

Question

Let $X$ be a Polish space with Borel probability measure $\mu$. A compact group $G$ acts on X continuously. It is right that for any $f\in C_b(X)$ exists a sequence $(g_k \in G)_{k\in \mathbb{N}}$ such that $$ \int_G (f\circ g)d\chi_G(g) = \lim_{n \rightarrow \infty }\frac{1}{n}\sum_{k=1}^n f\circ g_k $$ in the sense of $L_1(X,\mu)$ norm?