# Measurable sets in pratice with Lebesgue-Stieltjes measure

If we consider $$F \colon \mathbb{R} \to \mathbb{R},$$ defined as

$$F(x) = \left\{\begin{array}{cc} 0, & \mbox{if } x < 0 \\ 3, & \mbox{if } 0 \le x < 4 \\ 8, &\mbox{otherwise.} \end{array}\right.,$$

and let $$\mu_F$$ the corresponding Lebesgue-Stieltjes measure. What are the measurable sets in this measure? That is, the sets $$E$$ satisfying

$$\mu_F(A) = \mu_F(A \cap E) + \mu_F(A \setminus E)$$

$$\forall A \in \mathbb{R}.$$

$$\mu_F(A)=3\delta_0(A)+4\delta_4(A)$$ where $$\delta_x(A)=1$$ if $$x \in A$$ and $$0$$ otherwise. Since $$\delta_x(A)= \delta_x(A\cap E)+\delta_x(A\setminus E)$$ for all $$A,E \subset \mathbb R$$ it follows that all sets are measurable for $$\mu_F$$.