# What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability?

What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability?

I know Borel measurability implies both Lebesgue-Stieltjes measurability and Lebesgue measurability.

Does Lebesgue measurability implies Lebesgue-Stieltjes measurability, or the other? Or does it depend on the function corresponding to Lebesgue-Stieltjes measurability?

Def. Let $f$ be finite and monotone increasing on the real line. Define $\lambda(a,b]=f(b)-f(a)$. If $A$ is a nonempty subset of the real line, let $$\Lambda^*_f (A) = \inf \lambda(a_k,b_k],$$ where the inf is taken over all countable collections $\{ (a_k, b_k] \}$ such that $A \subseteq \bigcup (a_k, b_k]$. Furthe define the empty set measures 0. And by this outer measure construct the Lebesgue-Stieltjes measure $\Lambda_f$ on $\Lambda^*$ measurable sets.

• What is the definition of the LS-measurability? – Ilya Dec 4 '14 at 14:54