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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

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

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

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$\mu_F(A)=3\delta_0(A)+4\delta_4(A)$ 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 $\mu_F$.

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