Let $(X,\tau,\circ)$ be a compact Polish group. Is there necessarily a metric $d$ on $X$ inducing $\tau$ such that $d(x \circ a,x \circ b) = d(a \circ x, b \circ x) = d(a,b)$ for all $x,a,b \in X$?


If the answer is yes, I would appreciate a sketch of the argument establishing this.



Bi-invariant metrics on compact Polish group
Yes - the existence of a left-invariant metric on any Hausdorff first countable group is the Birkhoff-Kakutani theorem. Given a left-invariant metric $d$ on a compact group then just put 
$$
d'(a,b)=\sup\_x d(ax,bx) \;.
$$


PS Of course, the point is that all these metrics metrize the group topology.


