# Find a function sarting with a Lebesgue-Stieltjes measure

Exercise. Find a non-decreasing and right-continuous function $$F\colon\Bbb{R}\to\Bbb{R}$$ whose Lebesgue-Stieltjes associated $$\mu_F$$ satisfies the following conditions at same time

• $$\mu_F(\{0\})=\frac{1}{2}.$$

• $$\mu_F([0,\frac{1}{2}])=2.$$

• $$\mu_F((-\infty,-\frac{1}{2}])=0.$$

• $$\mu_F([1,x))=2x,\forall x>1.$$

I was thinking if I can use the following Theorem to solve this question. We denotes $$B_{\Bbb{R}}$$ the $$\sigma$$-algebra of Borel.

Theorem. Let $$F\colon\Bbb{R}\to\Bbb{R}$$ increasing and right-continuous function. So there is a unique measure $$\mu_F$$ defined in $$B_\Bbb{R}$$ such that $$\mu_F((a,b])=F(b)-F(a).$$

• If $$G\colon\Bbb{R}\to\Bbb{R}$$ is another increasing and right-continuous function, then $$\mu_F=\mu_G\Longleftrightarrow F-G=c,\ \text{where}\ c\in\Bbb{R}\ \text{constant}.$$

• Conversely, if $$\mu$$ is a measure defined in $$B_\Bbb{R}$$ that is finite for any bounded set, then $$F\colon\Bbb{R}\to\Bbb{R}$$ given by $$F(x)=\left\lbrace\begin{array}{c}\mu((0,x]),x>0\\ 0,x=0\\ -\mu((x,0]),x<0\end{array}\right.$$ is increasing and right-continuous and $$\mu_F=\mu$$.

Take $$F(x)=0$$ for $$x<0$$, $$\frac 1 2$$ for $$0\leq x <\frac 1 4$$, $$\frac 3 2$$ for $$\frac 1 4 \leq x <1$$ and $$2x$$ for $$x \geq 1$$. Apply the theorem to get $$\mu_F$$.