I can't figure out how to show that Lebesque-Stieljes measure is unique.

[sto]

I can't figure out how to show that Lebesque-Stieljes measure is unique.

I constructed the measure in the following manner: let g be a finite,
increasing, and continuous function of a real variable and define by m(
[a,b) ) = g(b) - g(a) a set function m on the class P of all sets of the
form {x:infty < a <= x < b < infty for some a,b real}. Then the class P
is a semiring, and m is non-negative, countably additive, with m(0)=0 on
P, in which case m has a unique extension to a measure on the ring R
generated by the semiring P. Furthermore, because m is finite on P the
extension of m to a measure on R is also finite (and therefore
sigma-finite).

Next for any E in the hereditary sigma-ring H(R) generated by R, define
the set function

m*(E) = inf {sum(i,infty) m(E_i) : E subset union(i,infty) E_i, E_i in R}

Then m* extends the measure m on R to a sigma-finite outer measure on
H(R). Furthermore, restricting m* to the class S_ of of all
m*-measurable sets induces a complete measure m_ on S_ (obviously
defined by m_(E) = m*(E)). Because m on R is sigma-finite, the measure
m_ on S_ is also sigma-finite. Furthermore, the class S_ contains the
sigma-ring S of Borel sets. We conclude that m_ on S_ is a complete
sigma-finite measure on a sigma-ring that contains the Borel sets.

I can also prove in two different ways that given any set E in S_, there
exists a Borel set F in S such that m_( symmetric_difference(E,F) ) = 0,
which I suspect in the key to the uniqueness on m_. (Briefly, one way is
by noting that any E in S_ has a measurable cover F in S, and then
calculating m_(F-E), and another way is by noting that m_ on S_ equals
the completion of the extension of m to S so that any E in S_ can be
written as the symmetric difference of a Borel set F in S and a subset N
of a null set G in S).

I've tried a number of relations at random, but I can't quite construct
an exact proof that m_ on S_ is unique. It never quite works out. Can
anyone give a proof?

Thanks,
-sto

[smn]

Any set in S_ is the disjoint union of a Borel set and a subset of a
borel set of m_ measure 0 .Any other positive measure which agrees
with m_ on Borel sets must give 0 on all subsets of Borel sets of
measure 0 by monotonicity of positive measures.Thus the 2 measures
must agree on the above disjoint union and hence on all of S_. smn

[sto]

smn wrote:
the completion of the extension of m to S so that any E in S_ can be
> written as the symmetric difference of a Borel set F in S and a subset N
> of a null set G in S).
>
>

> Any set in S_ is the disjoint union of a Borel set and a subset of a

This is where I got stuck. I did not notice that the union was
disjoint, which should have been obvious from

symmetric_difference(F,N) =
(F-G) union (G intersection [symmetric_difference(F,N)])

THanks,
-sto