Proof
For
we
calculate
We make the change
in the first
integral.
It follows from (1) that
is a linear operator in
and we conclude from the definition of
that
Hence, for
Therefore
thus
In view of the proposition
(
Properties of Feller resolvent
1
)3, it remains to prove that
is dense in
.
We
evaluate
for large
and aim to show that the quantity is small. We make a change
in the integral,
,
The quantity
is positive and tends to 0 when
tends to
.
Hence, using
(2),
In
addition
Therefore, by the proposition
(
Dominated convergence theorem
),
for any finite measure
Hence, if the measure
vanishes
on
then it vanishes on
.
Thus,
is dense in
.
Proposition
(Properties of Feller resolvent
2) Let
be a resolvent of a Feller process. We
have
