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Restrict $\mu$ first to loops surrounding $0$. This satisfies the pointed restriction property, so for one constant $C$,A single rooted pushforward cannot equal the whole ${\mu }_D$: it only sees loops surrounding its root. The rest is recovered by moving that root.

For $z\in D$, choose a disc automorphism ${\psi }_z$ with ${\psi }_z(0)=z$. Conformal restriction yieldswith the same constant. Let $(z_j)$ be a countable dense set in $D$. Every simple loop contained in $D$ surrounds one of these points. Make this cover disjoint by $P_j=E_{z_j}\setminus {\bigcup }_{i<j}{E}_{z_i}$. For a measurable event $A$ of loops in $D$,This explicitly relates the entire loop measure to the single-root measure without overcounting a loop once for every point in its interior. It also proves that knowledge of $\rho$, up to normalization, determines ${\mu }_D$. Nontriviality gives $C>0$: otherwise all countably many pointed patches would vanish. Using the usual unrooted Brownian loop measure, the same construction is commonly expressed as its outer-boundary pushforward, up to a constant.

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.

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.