Let W be a Borel subset of
I
×
I
I \times I
(where
I
=
[
0
,
1
]
I = [0,1]
) such that, for each x,
W
x
=
{
y
:
(
x
,
y
)
∈
W
}
{W_x} = \{ y:\,(x,y) \in W\}
is uncountable. It is shown that there is a map, g, of
I
×
I
I \times I
onto W such that (1) for each x,
g
(
x
,
⋅
)
g(x, \cdot )
is a Borel isomorphism of I onto
W
x
{W_x}
and (2) both g and
g
−
1
{g^{ - 1}}
are
S
(
I
×
I
)
S(I \times I)
-measurable maps. Here, if X is a topological space,
S
(
X
)
S(X)
is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that
S
(
X
)
S(X)
is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone. This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann. We also show an analogous result holds if W is only assumed to be analytic.