On page 13 of Bekka and Mayer "Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces" the authors study measure preserving actions of a locally compact topological group $G$ on a probability space $(X, \mu)$. They claim without further comment that given any $f \in L^2(X)$, the map $G \to L^2(X)$ given by $g \mapsto T_g f$ where $T_g f(x) = f(g^{-1} x)$ is continuous. This is not clear to me.
After thinking about it for a while, I think I can prove it if I can prove the seemingly simpler statement that given any measurable set $A$ of $X$, the function $h(g) = \mu(A \cap g.A)$ is a continuous function on $G$. However, I was unable to prove this statement, even in the special case of measure preserving $\mathbb{R}$ actions. I feel like there must be something that I'm missing. I also feel like this must be a standard result, but my Googling attempts were fruitless.
Hint: The $L^2$-norm is an integral. Use Lebesgue's Dominated Convergence Theorem to show what you want.