Let
P
1
,
…
,
P
d
{P_1}, \ldots ,{P_d}
be commuting positive linear contractions on
L
1
{L_1}
and let
T
1
,
…
,
T
d
{T_1}, \ldots ,{T_d}
be (not necessarily commuting) linear contractions on
L
1
{L_1}
such that
|
T
i
f
|
≤
P
i
|
f
|
|{T_i}f| \leq {P_i}|f|
for
1
≤
i
≤
d
1 \leq i \leq d
and
f
∈
L
1
f \in {L_1}
. In this paper we prove that if each
P
i
,
1
≤
i
≤
d
{P_i},1 \leq i \leq d
, satisfies the mean ergodic theorem, then the averages
A
n
(
T
1
,
…
,
T
d
)
f
=
A
n
(
T
1
)
⋯
A
n
(
T
d
)
f
{A_n}({T_1}, \ldots ,{T_d})f = {A_n}({T_1}) \cdots {A_n}({T_d})f
, where
A
n
(
T
i
)
=
n
−
1
∑
k
=
0
n
−
1
T
i
k
{A_n}({T_i}) = {n^{ - 1}}\sum \nolimits _{k = 0}^{n - 1} {T_i^k}
, converge a.e. for every
f
∈
L
1
f \in {L_1}
. When
T
1
,
…
,
T
d
{T_1}, \ldots ,{T_d}
commute, we further prove that the
L
1
{L_1}
-norm convergence of the averages
A
n
(
P
1
,
…
,
P
d
)
f
{A_n}({P_1}, \ldots ,{P_d})f
for every
f
∈
L
1
f \in {L_1}
implies the a.e. convergence of the averages
A
n
(
T
1
,
…
,
T
d
)
f
{A_n}({T_1}, \ldots ,{T_d})f
for every
f
∈
L
1
f \in {L_1}
. These improve Çömez and Lin’s ergodic theorem.