Let
R
\mathbb {R}
be the set of real numbers, and define
R
∞
=
∏
i
=
1
∞
R
\mathbb {R}^{\infty }=\prod \limits ^{\infty }_{i=1}\mathbb {R}
. We construct a complete measure space
(
R
∞
,
L
,
λ
)
(\mathbb {R}^{\infty },\mathcal {L},\lambda )
where the
σ
\sigma
-algebra
L
\mathcal {L}
contains the Borel subsets of
R
∞
\mathbb {R}^{\infty }
, and
λ
\lambda
is a translation-invariant measure such that for any measurable rectangle
R
=
∏
i
=
1
∞
R
i
R=\prod \limits ^{\infty }_{i=1}R_{i}
, if
0
≤
∏
i
=
1
∞
m
(
R
i
)
>
+
∞
0\le \prod \limits ^{\infty }_{i=1}m(R_{i})>+\infty
, then
λ
(
R
)
=
∏
i
=
1
∞
m
(
R
i
)
\lambda (R)=\prod \limits ^{\infty }_{i=1}m(R_{i})
, where
m
m
is Lebesgue measure on
R
\mathbb {R}
. The measure
λ
\lambda
is not
σ
\sigma
-finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure
λ
\lambda
, we construct, via selfadjoint operators on
L
2
(
R
∞
,
L
,
λ
)
L_{2}(\mathbb {R}^{\infty },\mathcal {L},\lambda )
, a “Schrödinger model” of the canonical commutation relations:
[
P
j
,
P
k
]
=
[
Q
j
,
Q
k
]
=
0
[P_{j},P_{k}]=[Q_{j},Q_{k}]=0
,
[
P
j
,
Q
k
]
=
i
δ
j
k
[P_{j},Q_{k}]=i\delta _{jk}
,
1
≤
j
,
k
>
+
∞
1\le j,k>+\infty
.