The proof is essentially just a formalization of
Kac's [33] original argument for space-time BM.
Variations appear in the proofs of similar results in
[38, 50, 51, 19].
Let Y denote the killed process with state space 
defined by 
if t<T and
if 
, where 
is a cemetery state.
In terms of Y, the killing time T is
just the hitting time of 
.
It is therefore enough to prove the
result for T the hitting time of a point in the state space E of X.
We assume, without any real loss of generality, that the sample space
is equipped with a family of shift operators 
, such that 
for all 
.
Furthermore, we assume that there is a filtration 
on
to which X is adapted and with respect to which X
has the simple Markov property:
for all 
, all
non-negative
-measurable functions F, and all
non-negative functions 
on that are measurable
with respect to 
.
The key property of a hitting time T of X is that it is a
terminal time [8, 59];
that is, an 
-stopping time T with the property
on the event 
.
The basic inductive step which allows Kac's moment formula
(4) to be pushed from n to n+1 involves the following identity,
which holds for an arbitrary -stopping time T.
Let 
be the pre-T occupation kernel defined by
for an arbitrary
non-negative measurable v.
If 
is an initial distribution, then 
is the measure
which describes the 
expected
amount of time X spends in various subsets F of E up to time T.
Call 
the pre-T occupation measure for
.
In case T is a Markov killing time of X, G is the potential
kernel derived from the killed process, as discussed in Section 1. But the
above
definition
(11) of
G makes sense, and the following identity is valid, for an arbitrary
stopping time T:
Occupation measure identity [38, 50, 51]
For each initial distribution on E,
each
non-negative 
-measurable ,
and each non-negative 
-measurable f,
The assumed measurability of 
implies that
is 
-measurable,
because is -measurable. Thus the left side of (13)
is well defined.
Now Fubini's theorem shows that
which by the Markov property (10) and the -measurability of
is equal to
Taken together, (12), (14), and (15) yield (13).
Notice that the proof of (13) required no strong Markov property of
X. So the occupation measure identity holds without any assumptions about the
state space of X or path properties of X beyond the joint measurability of
as a function of t and 
.
Turning to the proof of (4), observe that
where
Because T is a terminal time, the obvious change of variables in (17) leads to
Thus (4) follows by induction from the occupation measure identity (13).
