Step 1: We build a large separable space (Z, d Z ) equipped with a 1-Lipschitz map H, in which we have the convergence of Stump h k (T k ) to Stump h (T ) ,

As ?(C) is a class of R n and ?(?(C)) = ?(C), we have under our assumptions that ?(C) = C. This means that for all x ? F, n ? N * we have xR n ?(x), i.e. d(x, ?(x)) ? 2 n . It follows that for every x ? F , ?(x) = x. Now ,

, We can deduce that a.s. for every k ? N * , the sequence (Slice r k (T n )) n?N * is Cauchy in (X 0,K , d GHP ). When this is true, then by Lemma 3.3.9, (Slice r k (T n )) n?N * converges in (X S , d LGHP ) to a random measured labelled spaceT I . Since T and X 0,S are closed in X S ,T I is a random tree with null measure. The a.s. convergence of (T n ) n?N * toT in this coupling implies the convergence of their laws for the Prokhorov distance. The sequence (T n ) n?N * converges in law toT

, Since the trees defined by (5.3.3) (withT (n) instead ofT I for some large n) satisfy C4-5, it is reasonable to conjecture that the trees defined by (5? N * , and consider the tree (T (n) , (? h i , 1 ? i ? k)) as an element of T [k] which is still a Polish space, similar arguments as in the previous Section gives a limiting tree endowed with k measures (T I , (? h i , 1 ? i ? k)). We could make rigorous the construction of (T I , (? h ) h?E ) using the last part of the Remark 4.4.1 and considering it as a T [?] -valued random variable. We shall not provide of proof of this fact, but simply conjecture its existence, particular, this implies of I. The T

, Xn are probability measures) and the previous inequalities, we get Then, we believe that this regularity result is a corner stone to extend by continuity the family of probability measure (? h ) h?E to a family (? h ) h?I with for a.e. h that H? h = ? h , see Properties (5.2.1) and (5.2.2). Then Property (5.2.3) can be seen as a definition of ? I

, Let µ be a measure satisfying conditions C1-3. Then, for all h ? I, S h = (Stump h (T I ), (? h ) h ?(??,h] ) and C h = (Crown(h, T I )

, Since Stump h (T I ) is the Local-Gromov-Hausdorff limit of Stump h (T I ) when h ? E ? (??, h] goes to h ? and ? h is conjectured to be a measurable function of (? h ) h ?E?(??,h) , so we can express S h as a measurable function of (S h ) h ?E?(??,h) which is independent from C h

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