Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 5 / B1 / Solution

The restriction and normalization make $\mu$ the unique conformal-restriction loop measure surrounding the origin, up to the fixed multiplicative normalization. By the classification of planar conformal restriction measures, it is the image of the rooted Brownian loop measure at the origin under the map taking a Brownian loop to its outer boundary. It is scale invariant and sigma-finite, with finite mass after imposing lower and upper diameter cutoffs.