Let $([0,1],\mathcal{L},\lambda)$ be a measure space, where $\mathcal{L}$ is the completion the the Borel-$\sigma$-algebra on $[0,1]$ and $\lambda$ is the Lebesgue-measure. Let $\nu$ be any finitely additive finite set function on $\mathcal{L}$ which is 0 if $\lambda$ is. Note that I don't write absolute continuous, because this property is mostly defined for measures and we don't know if $\nu$ is $\sigma$-additive or positive.
Is the function $h:t \mapsto \nu([0,t])$ measurable? I tried to argue that it can be decomposed into positive parts and then using that they are increasing. But the Jordan decomposition needs signed measures. Thanks in advance.
You can use the Jordan decomposition for finitely additive set functions, see, e.g., decomposition of finitely additive measure