190 5. Exact Groups and Related TopicsNote that for every x, y EK and k, l, we have
{z E aK: y E 8(x,z,l,k)}
= LJ{T(x', y) : x' EK with d(x', x) ::; k and d(x', y) = l}
and hence the former set is Borel ink. Also, note the inclusion 8(x, z, l, k) c
8(x, z, l, k') whenever k ::; k'.
Let C = C(K) > 0 be the constant appearing in Lemma 5.3.8. Since K is
uniformly locally finite, there exists D > 0 such that every ball in K of radius
C contains at most D/3 points. By Lemma 5.3.8, the subset 8(x,z,l,k) is
contained in the C-tubular neighborhood of a subpath a([l - k, l + k]) of
any geodesic path a connecting x to z. This implies that l8(x, z, l, k)I ::; Dk
for all x, z, k, l with l ::?: k. For a finite subset 8 C K, we denote by xs E
Prob(K) the normalized characteristic function on 8. Define a sequence of
Borel functions T/n: K x f)K--+ Prob(K) by
l 2n
T/n(x, z) =;, L XS(x,z,3n,k)·
k=n+l
We claim that, for each x, x' E K, we have
lim sup llTJn(x, z) - TJn(x', z) 11 = 0.
n-->oo zEBKLet d = d(x, x'). Fix z E 8K and n ::?: d and set 8k = 8(x, z, 3n, k) and
8£ = 8(x', z, 3n, k). Then, we have 8k U 8£ c 8k+d and 8k n 8£ ~ 8k-d for
every n < k ::; 2n. It follows that
llxsk - Xs~ll = 2 ( 1-m~j~ 1 ~~~£I}) ::; 2 ( 1 - :~:::)
for n < k::; 2n. Since l8kl ::; Dk, we have
l 2n
llTJn(x, z) - TJn(x', z) II ::; - L llxsk - Xs;)
n k=n+l