4.3. Amenable actions 127Definition 4.3.5. An action of r on a compact space Xis called (topolog-
ically) amenable (or, equivalently, X is an amenable f-space) if there exists
a net of continuous maps mi: X-+ Prob(f), such that for each s Er,_lim (sup lls.mf - mf'xll1) = 0,
i--+oo xEX
where s.mf(g) = mf (s-^1 g).^6
Remark 4.3.6. Let Prob(X) be the set of all regular Borel probability
measures on X. In Proposition 5.2.1 we will show for a countable group r
that amenability can be reformulated as: For any finite subset E c r, s > 0
and any m E Prob(X), there exists a Borel mapμ: X-+ Prob(f) (i.e., the
function X-+ IR, x f---t μx(t), is Borel for every t Er) such thatmax r lls.μx - μs.x111 dm(x) < E.
sEE JxLemma 4.3. 7. An action a: r -+ Homeo(X) is amenable if and only if the
induced action on C(X) is amenable in the sense of Definition 4.3.1.Proof. The proofs of both directions are similar. First assume the action
is amenable in the sense of Definition 4.3.5. Let mi: X -+ Prob(r) be a
sequence of continuous maps such that for each s E r,
_lim (sup lls.mf - mf'xll1) = 0.
i--+oo xEX
Define Si: r-+ C(X) by
Si(g)(x) = mf(g).
Then for each x E X we have
L:si(g)(x) = I:mf(g) = 1,
g g
since mf is a probability measure. Defining Ti(g) = v'SJ:ij, it follows that
for each i,
(Ti,Ti) - - = ""'-L.JTi(g)^2 = lc(X)·^7
g
Of course, the Ti's are not finitely supported (we will fix that later) but note
that for each x E X,
(s *a Ti)(g)(x) = as('.fi(s-^1 g))(x) = Ti(s-^1 g)(s-^1 .x) = Js.mr^1 ·x(g).
6By definition, Prob(r) is the set of probability measures on r -which we identify with the
set of positive, norm-one elements in £^1 (r). Continuity means with respect to the restriction of
the weak-* topology on £^1 (r). In other words, m: X --+ Prob(r) is continuous if and only if for
each convergent net Xi --+ x EX we have m"'i (g) --+ m"'(g) for all g Er.
7Note that Dini's Theorem implies this sum converges uniformly, since everything is positive.