Constructing measure using a non-monotonic function

Question

It's well-known that we can construct a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ using a Stieltjes measure function $F:\mathbb{R} \to \mathbb{R}$ which is a non-decreasing right continuous function (see for example Durret Theorem 1.1.4). The steps are as follows:

1. Consider the semi-algebra $\mathcal{S}=\{(a,b]\cap \mathbb{R}: -\infty\le a\le b \le \infty\}$ and define $\mu((a,b]\cap \mathbb{R}) = F(b)-F(a)$. By the telescoping series argument, it can be shown that $\mu$ is finitely additive on $\mathcal{S}$.
2. Using the right continuity of $F$ and Heine-Borel theorem, it can be sown that $\mu$ is countably sub-additive on $\mathcal{S}$.
3. Now extension theorems can be invoked in order to extend the measure $\mu$ to the algebra $\mathcal{A}(\mathcal{S})$ and then (the resulsting measure is $\sigma$-finite) to the sigma-algebra $\sigma(\mathcal{S})$.

I think the monotonicity of $F$ is only required for $\lim_{x \to \pm\infty} F(x) = a \ \text{or} \pm\infty$ to be true. Is it possible to consider a non-monotonic function $F$ and only require the mentioned condition? I mean considering function $F$ such that $F$ is right continuous and $$\lim_{x \to \pm\infty} F(x) = a \ \text{or} \ \pm\infty.$$

Answer 1

If $F$ is a real function of bounded variation which is right continuous then there is a unique real measure $\mu$ on The Borel $\sigma-$ algebra of $\mathbb R$ such that $\mu ((-\infty, x])=F(x)$ for all $x$. In fact, we can even take $F$ to be complex valued, in which case $\mu$ becomes a complex measure. Further, $F$ is monotonically increasing if and only if $\mu$ is a positive measure.