Solution
= 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.