Let
G
G
be a locally compact group and
E
E
a complex Banach space. Let
φ
:
G
→
E
\varphi :G \to E
be a function which is the Fourier transform of a weakly compact operator
Φ
:
C
∗
(
G
)
→
E
\Phi :{C^*}(G) \to E
in the sense that
Φ
∗
∗
(
ω
(
s
)
)
=
ϕ
(
s
)
{\Phi ^{**}}(\omega (s)) = \phi (s)
,
s
∈
G
s \in G
, where
ω
:
G
→
W
∗
(
G
)
⊂
L
(
H
ω
)
\omega :G \to {W^*}(G) \subset L({H_\omega })
corresponds to the universal representation of
C
∗
(
G
)
{C^ * }(G)
. It is proved that
lim
i
∫
ϕ
d
μ
i
=
Φ
∗
∗
(
p
ω
)
{\lim _i}\smallint \phi d{\mu _i} = {\Phi ^{**}}({p_\omega })
, where
p
ω
{p_\omega }
is the projection onto the space of the common fixed points of all
ω
(
s
)
\omega (s)
,
s
∈
G
s \in G
, and
(
μ
i
)
i
∈
I
{({\mu _i})_{i \in \mathcal {I}}}
is an arbitrary net in the measure algebra
M
(
G
)
M(G)
satisfying
sup
i
∈
I
‖
ω
(
μ
i
)
‖
>
∞
{\sup _{i \in \mathcal {I}}}\left \| {\omega ({\mu _i})} \right \| > \infty
,
lim
i
μ
i
(
G
)
=
1
{\lim _i}{\mu _i}(G) = 1
, and
lim
i
‖
ω
(
μ
i
∗
∗
δ
s
−
μ
i
∗
)
ξ
‖
=
0
{\lim _i}\left \| {\omega (\mu _i^* * {\delta _s} - \mu _i^*)\xi } \right \| = 0
for all
s
∈
G
s \in G
,
ξ
∈
H
ω
\xi \in {H_\omega }
. If
E
E
is a Hilbert space and
ϕ
\phi
left (resp. right) homogeneous, the second (resp. first) of the last two limit conditions may be omitted. Finally, a connection of such random fields
ϕ
\phi
to a measurability condition is established.