AbstractAssume $$ (\Omega , {\mathscr {A}}, P) $$
(
Ω
,
A
,
P
)
is a probability space, X is a compact metric space with the $$ \sigma $$
σ
algebra $$ {\mathscr {B}} $$
B
of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$
f
:
X
×
Ω
→
X
is $$ {\mathscr {B}} \otimes {\mathscr {A}} $$
B
⊗
A
measurable and contractive in mean. We consider the sequence of iterates of f defined on $$ X \times \Omega ^{{\mathbb {N}}}$$
X
×
Ω
N
by $$f^0(x, \omega ) = x$$
f
0
(
x
,
ω
)
=
x
and $$ f^n(x, \omega ) = f\big (f^{n1}(x, \omega ), \omega _n\big )$$
f
n
(
x
,
ω
)
=
f
(
f
n

1
(
x
,
ω
)
,
ω
n
)
for $$n \in {\mathbb {N}}$$
n
∈
N
, and its weak limit $$\pi $$
π
. We show that if $$\psi :X \rightarrow {\mathbb {R}}$$
ψ
:
X
→
R
is continuous, then for every $$ x \in X $$
x
∈
X
the sequence $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$
1
n
∑
k
=
1
n
ψ
(
f
k
(
x
,
·
)
)
n
∈
N
converges almost surely to $$\int _X\psi d\pi $$
∫
X
ψ
d
π
. In fact, we are focusing on the case where the metric space is complete and separable.