How can we prove that Lebesgue-Stieltjes measures are regular Borel measures?

[hermanni]

I need to show that any lebesgue stieltjes measure is a regular borel measure. I'm really clueless , can anyone help??
We know the definition and facts about the distribution function , how can we conclude approximation by compact or closed sets??
Regards, hermanni.

 

[micromass]

First, can you give your definition of Lebesque-Stieltjes measure??

Let $$\mu$$ be the Lebesque-Stieltjes measure, we need to show that for every Borel set A holds that

$$\mu(A)=\inf\{\mu(G)~\vert~G~\text{open and}~A\subseteq G\}$$

and something analogous in the case of closed sets.
Can you prove this for A=]a,b]? Can you find a sequence of closed sets G_n containing ]a,b] such that $$\lim_{n\rightarrow +\infty}{\mu(G_n)}=\mu(]a,b])$$??


Of you've proven this first case, then you'll need to use that the intervals of the form ]a,b] form a semiring (i.e. apply an approximation theorem.)

 

[hermanni]

Ok , here's our course's definition : Let F be a right-continuos and nondecreasing function.
Then lebesgue - stieltjes measure associated to F is u and:
u(a, b] = F(b) - F(a)
For the compact sets , we do approximation from inside .The thing that bothers me is extension from a semiring to the ring.Any way , I'll try your suggestions , thanks

 

[hermanni]

Hi,
I showed approximations for intervals. Can you give me an idea how I can show it for any set??
Regards, hermanni.

 

[micromass]

You'll need an approximation theorem.
Have you seen the following?

If $$\mathcal{A}$$ is a semiring and if A is a Borel set. Then there exist $$A_1,...,A_n$$ such that

$$A\subseteq \bigcup A_i~\text{and}~\mu\left(\bigcup{A_i}\setminus A\right)<\epsilon$$

Or did you see any other theorem that looks like it?

 

[hermanni]

Actually no , in the course we only saw that if we have a premasure on a semiring , then we can extend it to a measure on the ring.
Also we noted down

If $$E \in S$$ and $$F \in S$$ then there exists a finite number of mutually disjoint sets $$C_i \in S$$ for $$i=1,\ldots,n$$ such that $$E \setminus F = \cup_{i=1}^n C_i$$ without proof , it looks like what you said.Can you explain how the result will follow from your lemma? We also did something similar in the class at characterization of the measurable sets: If A is any lebesgue measurable set , then what you said follows and Ai's are open sets.