Let T, X be Polish spaces,
J
\mathcal {J}
a countably generated sub-
σ
\sigma
-field of
B
T
{\mathcal {B}_T}
, the Borel
σ
\sigma
-field of T, and
F
:
T
→
X
F:\,T\, \to \,X
a multifunction such that
F
(
t
)
F(t)
is a
G
δ
{G_\delta }
in X for each
t
∈
T
t\, \in \,T
. F is
J
\mathcal {J}
-measurable and
Gr
(
F
)
∈
J
⊗
B
X
{\text {Gr}}(F)\, \in \,J\, \otimes \,{\mathcal {B}_X}
, where
Gr
(
F
)
{\text {Gr}}(F)
denotes the graph of F. We prove the following three results on F. (I) There is a map
f
:
T
×
Σ
→
X
f:\,T\, \times \,\Sigma \, \to \,X
such that for each
t
∈
T
,
f
(
t
,
⋅
)
t\, \in \,T,\,f(t,\, \cdot )
is a continuous, open map from
Σ
\Sigma
onto
F
(
t
)
F(t)
and for each
σ
∈
Σ
,
f
(
⋅
,
σ
)
\sigma \, \in \,\Sigma ,\,f( \cdot ,\,\sigma )
is
J
\mathcal {J}
-measurable, where
Σ
\Sigma
is the space of irrationals. (II) The multifunction F is of Souslin type. (III) If X is uncountable and
F
(
t
)
,
t
∈
T
F(t),\,t\, \in \,T
, are all dense-in-itself then there is a
J
⊗
B
X
\mathcal {J}\, \otimes \,{\mathcal {B} _X}
-measurable map
f
:
T
×
X
→
X
f:\,T\, \times \,X\, \to \,X
such that for each
t
∈
T
,
f
(
t
,
⋅
)
t\, \in \,T,\,f(t,\, \cdot )
is a Borel isomorphism of X onto
F
(
t
)
F(t)
.