= Finiteness propagation of Brownian mean exit times
{title2=$g_D(x)<\infty\text{ at one point}\Longrightarrow g_D<\infty\text{ throughout }D$}
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.
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