<s>
In	O
mathematics	O
,	O
cylinder	B-Algorithm
set	I-Algorithm
measure	I-Algorithm
(	O
or	O
promeasure	B-Algorithm
,	O
or	O
premeasure	O
,	O
or	O
quasi-measure	B-Algorithm
,	O
or	O
CSM	O
)	O
is	O
a	O
kind	O
of	O
prototype	O
for	O
a	O
measure	O
on	O
an	O
infinite-dimensional	O
vector	O
space	O
.	O
</s>
<s>
An	O
example	O
is	O
the	O
Gaussian	B-Algorithm
cylinder	I-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
Hilbert	O
space	O
.	O
</s>
<s>
Cylinder	B-Algorithm
set	I-Algorithm
measures	I-Algorithm
are	O
in	O
general	O
not	O
measures	O
(	O
and	O
in	O
particular	O
need	O
not	O
be	O
countably	O
additive	O
but	O
only	O
finitely	O
additive	O
)	O
,	O
but	O
can	O
be	O
used	O
to	O
define	O
measures	O
,	O
such	O
as	O
classical	O
Wiener	O
measure	O
on	O
the	O
set	O
of	O
continuous	O
paths	O
starting	O
at	O
the	O
origin	O
in	O
Euclidean	O
space	O
.	O
</s>
<s>
Let	O
be	O
a	O
separable	O
real	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
Let	O
denote	O
the	O
collection	O
of	O
all	O
surjective	B-Algorithm
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
defined	O
on	O
whose	O
image	O
is	O
some	O
finite-dimensional	O
real	O
vector	O
space	O
:	O
</s>
<s>
where	O
is	O
a	O
probability	O
measure	O
on	O
These	O
measures	O
are	O
required	O
to	O
satisfy	O
the	O
following	O
consistency	O
condition	O
:	O
if	O
is	O
a	O
surjective	B-Algorithm
projection	O
,	O
then	O
the	O
push	O
forward	O
of	O
the	O
measure	O
is	O
as	O
follows	O
:	O
</s>
<s>
is	O
modelled	O
on	O
the	O
way	O
that	O
true	O
measures	O
push	O
forward	O
(	O
see	O
the	O
section	O
cylinder	B-Algorithm
set	I-Algorithm
measures	I-Algorithm
versus	O
true	O
measures	O
)	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
important	O
to	O
understand	O
that	O
in	O
the	O
case	O
of	O
cylinder	B-Algorithm
set	I-Algorithm
measures	I-Algorithm
,	O
this	O
is	O
a	O
requirement	O
that	O
is	O
part	O
of	O
the	O
definition	O
,	O
not	O
a	O
result	O
.	O
</s>
<s>
The	O
canonical	O
Gaussian	B-Algorithm
cylinder	I-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
an	O
infinite-dimensional	O
separable	O
Hilbert	O
space	O
does	O
not	O
correspond	O
to	O
a	O
true	O
measure	O
on	O
The	O
proof	O
is	O
quite	O
simple	O
:	O
the	O
ball	O
of	O
radius	O
(	O
and	O
center	O
0	O
)	O
has	O
measure	O
at	O
most	O
equal	O
to	O
that	O
of	O
the	O
ball	O
of	O
radius	O
in	O
an	O
-dimensional	O
Hilbert	O
space	O
,	O
and	O
this	O
tends	O
to	O
0	O
as	O
tends	O
to	O
infinity	O
.	O
</s>
<s>
An	O
alternative	O
proof	O
that	O
the	O
Gaussian	B-Algorithm
cylinder	I-Algorithm
set	I-Algorithm
measure	I-Algorithm
is	O
not	O
a	O
measure	O
uses	O
the	O
Cameron	O
–	O
Martin	O
theorem	O
and	O
a	O
result	O
on	O
the	O
quasi-invariance	O
of	O
measures	O
.	O
</s>
<s>
Sazonov	O
's	O
theorem	O
gives	O
conditions	O
under	O
which	O
the	O
push	O
forward	O
of	O
a	O
canonical	O
Gaussian	B-Algorithm
cylinder	I-Algorithm
set	I-Algorithm
measure	I-Algorithm
can	O
be	O
turned	O
into	O
a	O
true	O
measure	O
.	O
</s>
<s>
A	O
cylinder	B-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
the	O
dual	O
of	O
a	O
nuclear	B-Algorithm
Fréchet	B-Algorithm
space	I-Algorithm
automatically	O
extends	O
to	O
a	O
measure	O
if	O
its	O
Fourier	O
transform	O
is	O
continuous	O
.	O
</s>
<s>
Example	O
:	O
Let	O
be	O
the	O
space	O
of	O
Schwartz	O
functions	O
on	O
a	O
finite	O
dimensional	O
vector	O
space	O
;	O
it	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
It	O
is	O
contained	O
in	O
the	O
Hilbert	O
space	O
of	O
functions	O
,	O
which	O
is	O
in	O
turn	O
contained	O
in	O
the	O
space	O
of	O
tempered	O
distributions	O
the	O
dual	O
of	O
the	O
nuclear	B-Algorithm
Fréchet	B-Algorithm
space	I-Algorithm
:	O
</s>
<s>
The	O
Gaussian	B-Algorithm
cylinder	I-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
gives	O
a	O
cylinder	B-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
the	O
space	O
of	O
tempered	O
distributions	O
,	O
which	O
extends	O
to	O
a	O
measure	O
on	O
the	O
space	O
of	O
tempered	O
distributions	O
,	O
</s>
