<s>
In	O
mathematics	O
,	O
the	O
Pettis	B-Algorithm
integral	I-Algorithm
or	O
Gelfand	B-Algorithm
–	I-Algorithm
Pettis	I-Algorithm
integral	I-Algorithm
,	O
named	O
after	O
Israel	O
M	O
.	O
Gelfand	O
and	O
Billy	O
James	O
Pettis	O
,	O
extends	O
the	O
definition	O
of	O
the	O
Lebesgue	O
integral	O
to	O
vector-valued	O
functions	O
on	O
a	O
measure	O
space	O
,	O
by	O
exploiting	O
duality	B-Algorithm
.	O
</s>
<s>
The	O
integral	O
is	O
also	O
called	O
the	O
weak	B-Algorithm
integral	I-Algorithm
in	O
contrast	O
to	O
the	O
Bochner	O
integral	O
,	O
which	O
is	O
the	O
strong	O
integral	O
.	O
</s>
<s>
Let	O
where	O
is	O
a	O
measure	O
space	O
and	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
a	O
continuous	O
dual	O
space	O
that	O
separates	O
points	O
(	O
that	O
is	O
,	O
if	O
is	O
nonzero	O
then	O
there	O
is	O
some	O
such	O
that	O
)	O
,	O
for	O
example	O
,	O
is	O
a	O
normed	O
space	O
or	O
(	O
more	O
generally	O
)	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
Evaluation	O
of	O
a	O
functional	O
may	O
be	O
written	O
as	O
a	O
duality	B-Algorithm
pairing	I-Algorithm
:	O
</s>
<s>
The	O
space	O
is	O
a	O
semi-reflexive	O
space	O
if	O
and	O
only	O
if	O
this	O
map	O
is	O
surjective	B-Algorithm
.	O
</s>
<s>
An	O
immediate	O
consequence	O
of	O
the	O
definition	O
is	O
that	O
Pettis	B-Algorithm
integrals	I-Algorithm
are	O
compatible	O
with	O
continuous	O
,	O
linear	O
operators	O
:	O
If	O
is	O
and	O
linear	O
and	O
continuous	O
and	O
is	O
Pettis	O
integrable	O
,	O
then	O
is	O
Pettis	O
integrable	O
as	O
well	O
and	O
:	O
</s>
<s>
The	O
standard	O
estimate	O
for	O
real	O
-	O
and	O
complex-valued	O
functions	O
generalises	O
to	O
Pettis	B-Algorithm
integrals	I-Algorithm
in	O
the	O
following	O
sense	O
:	O
For	O
all	O
continuous	O
seminorms	O
and	O
all	O
Pettis	O
integrable	O
holds	O
.	O
</s>
<s>
An	O
important	O
property	O
is	O
that	O
the	O
Pettis	B-Algorithm
integral	I-Algorithm
with	O
respect	O
to	O
a	O
finite	O
measure	O
is	O
contained	O
in	O
the	O
closure	O
of	O
the	O
convex	O
hull	O
of	O
the	O
values	O
scaled	O
by	O
the	O
measure	O
of	O
the	O
integration	O
domain	O
:	O
</s>
<s>
If	O
is	O
a	O
topological	O
space	O
,	O
its	O
Borel--algebra	O
,	O
a	O
Borel	O
measure	O
that	O
assigns	O
finite	O
values	O
to	O
compact	O
subsets	O
,	O
is	O
quasi-complete	B-Algorithm
(	O
that	O
is	O
,	O
every	O
bounded	O
Cauchy	O
net	O
converges	O
)	O
and	O
if	O
is	O
continuous	O
with	O
compact	O
support	O
,	O
then	O
is	O
Pettis	O
integrable	O
.	O
</s>
<s>
Let	O
be	O
a	O
probability	O
space	O
,	O
and	O
let	O
be	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
with	O
a	O
dual	O
space	O
that	O
separates	O
points	O
.	O
</s>
<s>
Let	O
be	O
a	O
sequence	O
of	O
Pettis-integrable	O
random	O
variables	O
,	O
and	O
write	O
for	O
the	O
Pettis	B-Algorithm
integral	I-Algorithm
of	O
(	O
over	O
)	O
.	O
</s>
