 Finiteness propagation of Brownian mean exit times

For a connected open set $D$, finiteness of the Brownian exit time expectation at one point implies finiteness everywhere. The Brownian exit-time ball averaging identity gives integrability on a ball centred at any finite point. Every point of that ball has a smaller centred ball contained in it; the same identity then gives a finite value there. The set of finite points is open. It is also relatively closed: near a limit point in $D$, choose a fixed ball radius smaller than a quarter of its distance to $D^c$ and centre it at a nearby finite point. The limit point lies in that ball and hence is finite. Connectedness completes the proof.