Uniform Hausdorff measure of the level sets of the Brownian tree

Let (T, d) be the random real tree with root rho coded by a Brownian excursion. So (T, d) is (up to normalisation) Aldous CRT Aldous (1991) (see Le Gall (1991). The a-level set of T is the set T(a) of all points in T that are at distance a from the root. We know from Duquesne and Le Gall (2006) that for any fixed a is an element of (0, infinity), the measure l(a) that is induced on T(a) by the local time at a of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in T with gauge function g(r) = r log log 1/r, restricted to T(a). As suggested by a result due to Perkins (1988, 1989) for super-Brownian motion, we prove in this paper a more precise statement that holds almost surely uniformly in a, and we specify the multiplicative constant. Namely, we prove that almost surely for any a is an element of (0, infinity), l(a)(.) = 1/2H(g)(. boolean AND T(a)), where H-g stands for the g-Hausdorff measure.