Let
(
X
,
M
,
μ
)
(X,\mathfrak {M},\mu )
be a finite measure space,
T
T
an invertible measure-preserving transformation and
υ
\upsilon
a positive measurable function. For
p
=
1
p = 1
, we prove that the ergodic Hubert transform
H
f
(
X
)
=
li
m
n
→
∞
∑
i
=
−
n
n
′
f
(
T
i
x
)
/
i
Hf(X) = {\text {li}}{{\text {m}}_{n \to \infty }}\sum \nolimits _{i = - n}^n {’f({T^i}x)/i}
exists a.e. for every
f
f
in
L
1
(
υ
d
μ
)
{L^1}(\upsilon d\mu )
if and only if
in
f
i
≥
0
υ
(
T
i
x
)
>
0
{\text {in}}{{\text {f}}_{i \geq 0}}\upsilon ({T^i}x) > 0
a.e. We also solve the problem for
1
>
p
≤
2
1 > p \leq 2
. In this case the condition is
su
p
k
≥
1
k
−
1
∑
i
−
0
k
−
1
υ
−
1
/
(
p
−
1
)
(
T
i
x
)
>
∞
{\text {su}}{{\text {p}}_{k \geq 1}}{k^{ - 1}}\sum \nolimits _{i - 0}^{k - 1} {{\upsilon ^{ - 1/(p - 1)}}} ({T^i}x) > \infty
a.e. If the transformation
T
T
is ergodic, the characterizing conditions become that
1
/
υ
∈
L
∞
1/\upsilon \in {L^\infty }
and
υ
−
1
/
(
p
−
1
)
∈
L
1
(
μ
)
{\upsilon ^{ - 1/(p - 1)}} \in {L^1}(\mu )
, respectively. These characterizations, together with some recent results, give, for
1
≤
p
≤
2
1 \leq p \leq 2
, that
H
f
(
x
)
Hf(x)
exists a.e. for every
f
f
in
L
p
(
υ
d
μ
)
{L^p}(\upsilon d\mu )
if and only if the sequence of the Césàro-averages
k
−
1
(
f
(
x
)
+
f
(
T
x
)
+
…
f
(
T
k
−
1
x
)
)
{k^{ - 1}}(f(x) + f(Tx) + \ldots f({T^{k - 1}}x))
converge a.e. for every
f
f
in
L
p
(
υ
d
μ
)
{L^p}(\upsilon d\mu )
. This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on
L
2
{L^2}
.