= Pinned Brownian loop measure at an interior point
{c}
{title2=$\rho\{B\not\subset U\}=-\log\operatorname{crad}(U,0)$}
In the unit disc, the relevant pointed <Brownian loop measure> is rooted at $0$ and normalized by its conformal deficit: for simply connected $U$ containing $0$, its mass of loops not contained in $U$ is minus the logarithm of the <conformal radius>. Its outer-boundary pushforward gives the corresponding pointed simple-loop restriction measure. It is not the full unrooted loop measure or a probability law of a fixed-duration Brownian bridge.
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