1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.2. Groups acting on trees 177

Proof. We first claim that for every E > 0 and finite subset E c r, there
exists a continuous map 'r/: X --+ Prob(K) such that

max sup [[s.fJx - 'r/s.x[[ < E.

sEE xEX

Let E c r be a fixed finite symmetric subset containing e. For every con-
tinuous map'(/: X--+ Prob(K), we define fry E C(X) by

fry(x) = L [Js.rJx - 'r/s.x[[ =LL [TJx(s-I.a) - 'r/s.x(a)[.

sEE sEEaEK

Observe that Jz:,k ak(k :S l:k akf (k for every ak 2: 0 with l:k ak = 1. Hence,
it suffices to show that 0 is in the norm-closed convex hull of {fry : 'r/: X --+
Prob(K) is continuous}. By the Hahn-Banach separation theorem, it ac-
tually suffices to show 0 is in the weak closure of this set. That is, by the
Riesz representation theorem, we must show that for every finite set of reg-
ular Borel probability measures μi, ... , μn on X, there exists a continuous
function 'r/: X --+ Prob(K) such that J fry dμi < E, for i = 1, ... , n.

Letting m = ~I: μi, a little thought reveals that we really only have to
find 'r/ such that J fry dm < E (for a smaller Ethan that above). So, let E > 0
be arbitrary. By assumption, there exists a Borel map (: X --+ Prob(K)
such that

I: r 11s.c -c·x11 dm(x) < ~-
sEE Jx
Fubini's Theorem and the fact that (s.x is a probability measure implies

1 = r (I: (^8 .x(a)) dm(x) =I: j c·x(a) dm(x),

lx aEK aEK X

for every s. Hence we can find a finite subset F C K such that

L 1 L (s.x(a) dm(x) < ~-

sEE X aEI'\F

By Lusin's Theorem (applied to the measure I:sEE s.m) we can approxi-
mate, for each a E F, the Borel function x ~ C (a) by a continuous function
x ~ TJx(a) so that

I: I: 11'(/s.x(a) - c·x(a)t dm(x) < ~·

sEEaEF X

Now fix ao E K \ F and define rJx(ao) = 1 - l:aEF rJx(a), for every x EX.
For b tj. FU {ao} we define rJx(b) = 0 for all x E X. We may assume that
TJx(ao) 2: 0 and regard 'r/ as a continuous map'(/: X--+ Prob(K) such that
supp 'r/x c F U { a 0 } for all x E X. It follows that

I: r llTJs.x - (s.x11 dm(x) < ~.

sEEJx