We construct a translation invariant Borel measure
λ
\lambda
on
R
∞
=
∏
i
=
1
∞
R
{{\mathbf {R}}^\infty } = \prod _{i = 1}^\infty {\mathbf {R}}
such that for any infinite-dimensional rectangle
R
=
∏
i
=
1
∞
(
a
i
,
b
i
)
,
−
∞
>
a
i
≤
b
i
>
+
∞
R = \prod _{i = 1}^\infty ({a_i},{b_i}), - \infty > {a_i} \leq {b_i} > + \infty
, if
0
≤
∏
i
=
1
∞
(
b
i
−
a
i
)
>
+
∞
0 \leq \prod _{i = 1}^\infty ({b_i} - {a_i}) > + \infty
, then
λ
(
R
)
=
∏
i
=
1
∞
(
b
i
−
a
i
)
\lambda (R) = \prod _{i = 1}^\infty ({b_i} - {a_i})
. Because
R
∞
{{\mathbf {R}}^\infty }
is an infinite-dimensional locally convex topological vector space, the measure
λ
\lambda
can not be
σ
\sigma
-finite.