<s>
The	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
,	O
named	O
for	O
Pierre-Simon	O
Laplace	O
and	O
Thomas	O
Joannes	O
Stieltjes	O
,	O
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
similar	O
to	O
the	O
Laplace	O
transform	O
.	O
</s>
<s>
It	O
is	O
useful	O
in	O
a	O
number	O
of	O
areas	O
of	O
mathematics	O
,	O
including	O
functional	B-Application
analysis	I-Application
,	O
and	O
certain	O
areas	O
of	O
theoretical	O
and	O
applied	O
probability	O
.	O
</s>
<s>
The	O
unilateral	O
(	O
one-sided	O
)	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
is	O
given	O
by	O
The	O
limit	O
is	O
necessary	O
to	O
ensure	O
the	O
transform	O
captures	O
a	O
possible	O
jump	O
in	O
at	O
,	O
as	O
is	O
needed	O
to	O
make	O
sense	O
of	O
the	O
Laplace	O
transform	O
of	O
the	O
Dirac	O
delta	O
function	O
.	O
</s>
<s>
The	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
in	O
the	O
case	O
of	O
a	O
scalar-valued	O
function	O
is	O
thus	O
seen	O
to	O
be	O
a	O
special	O
case	O
of	O
the	O
Laplace	O
transform	O
of	O
a	O
Stieltjes	O
measure	O
.	O
</s>
<s>
The	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
appears	O
naturally	O
in	O
the	O
following	O
context	O
.	O
</s>
<s>
If	O
X	O
is	O
a	O
random	O
variable	O
with	O
cumulative	O
distribution	O
function	O
F	O
,	O
then	O
the	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
is	O
given	O
by	O
the	O
expectation	O
:	O
</s>
<s>
The	O
Laplace-Stieltjes	B-Algorithm
transform	I-Algorithm
of	O
a	O
real	O
random	O
variable	O
's	O
cumulative	O
distribution	O
function	O
is	O
therefore	O
equal	O
to	O
the	O
random	O
variable	O
's	O
moment-generating	O
function	O
,	O
but	O
with	O
the	O
sign	O
of	O
the	O
argument	O
reversed	O
.	O
</s>
<s>
Whereas	O
the	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
of	O
a	O
real-valued	O
function	O
is	O
a	O
special	O
case	O
of	O
the	O
Laplace	O
transform	O
of	O
a	O
measure	O
applied	O
to	O
the	O
associated	O
Stieltjes	O
measure	O
,	O
the	O
conventional	O
Laplace	O
transform	O
cannot	O
handle	O
vector	O
measures	O
:	O
measures	O
with	O
values	O
in	O
a	O
Banach	O
space	O
.	O
</s>
<s>
exists	O
,	O
then	O
the	O
value	O
of	O
this	O
limit	O
is	O
the	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
of	O
g	O
.	O
</s>
<s>
The	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
is	O
closely	O
related	O
to	O
other	O
integral	B-Algorithm
transforms	I-Algorithm
,	O
including	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
the	O
Laplace	O
transform	O
.	O
</s>
<s>
If	O
g	O
has	O
derivative	O
g	O
 '	O
then	O
the	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
of	O
g	O
is	O
the	O
Laplace	O
transform	O
of	O
g′	O
.	O
</s>
