= Brownian loop measure
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{title2=$\mu^{\rm loop}$}
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Brownian loop measure is a <sigma-finite measure> on unrooted planar <Brownian motion> loops. A loop has a duration $t>0$; changing its starting time cyclically represents the same unrooted loop. Let $\mathbb P^{\rm br}_{z,t}$ be the probability law of a planar <Brownian bridge> from $z$ to $z$ of duration $t$. For a nonnegative measurable function $F$ of an unrooted loop,
$$
\mu^{\rm loop}(F)=\int_{\mathbb C}\int_0^\infty\frac{\mathbb E^{\rm br}_{z,t}[F]}{2\pi t^2}\,dt\,dA(z).
$$
Here $dA$ is planar <Lebesgue measure>. The weight combines the planar <Brownian transition density> $1/(2\pi t)$ with a further factor $1/t$ for unrooting. Restricting to loops contained in a domain gives its loop measure. A <conformal map>, with the <conformal Brownian clock> on each loop, transports this restricted measure to that of the image domain.<The Brownian loop soup by Gregory F. Lawler and Wendelin Werner>[Section 4.1, equation (15) and Proposition 6]{ref}
A <Poisson point process> with intensity $\lambda\mu^{\rm loop}$ is a <Brownian loop soup>. This construction connects <Brownian motion> to <conformal restriction measure on simple loops> through loop boundaries. The measure is infinite, so drawing one loop requires a finite restriction and normalization.
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