Stieltjes measure (1). (i) Let (mu) be a measure on ((mathbb{R}, mathscr{B}(mathbb{R}))) such that (mu[-n, n) 0

Stieltjes measure (1).

(i) Let \(\mu\) be a measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) such that \(\mu[-n, n)<\infty\) for all \(n \in \mathbb{N}\). Show that

\[F_{\mu}(x):= \begin{cases}\mu[0, x) & \text { if } x>0 \\ 0 & \text { if } x=0 \\ -\mu[x, 0) & \text { if } x<0\end{cases}\]

is a monotonically increasing and left-continuous function \(F_{\mu}: \mathbb{R} ightarrow \mathbb{R}\).

Remark. Increasing and left-continuous functions are called Stieltjes functions.

(ii) Let \(F: \mathbb{R} ightarrow \mathbb{R}\) be a Stieltjes function (see part (i)). Show that

\[u_{F}([a, b)):=F(b)-F(a), \quad \forall a, b \in \mathbb{R}, a

has a unique extension to a measure on \(\mathscr{B}(\mathbb{R})\).

[ check the assumptions of Theorem 6.1 with \(\mathcal{S}=\{[a, b): a \leqslant b\}\).]

(iii) Conclude that for every measure \(\mu\) on \((\mathbb{R}, \mathscr{B}(\mathbb{R})\) ) satisfying \(\mu[-r, r)<\infty, r>0\), there is some Stieltjes function \(F=F_{\mu}\) such that \(\mu=u_{F}\) (as in part (ii)).

(iv) Which Stieltjes function \(F\) corresponds to \(\lambda\) (one-dimensional Lebesgue measure)?

(v) Which Stieltjes function \(F\) corresponds to \(\delta_{0}\) (Dirac measure on \(\mathbb{R}\) )?

(vi) Show that \(F_{\mu}\) as in part (i) is continuous at \(x \in \mathbb{R}\) if, and only if, \(\mu\{x\}=0\).