Let $f:\mathbb R\to\mathbb R$ be a continuous increasing function. Define the (generalized) length of (finite) semiopen intervals,
$ \begin{align} \lambda_f:&\{[a,b):a,b\in\mathbb R\,;\;a\leq b\}\to[0,\infty),\\ &[a,b) \mapsto f(b)-f(a). \end{align} $
Define also
$ \begin{align}\theta^*_f:&\mathcal P(\mathbb R)\to[0,\infty],\\ &A\mapsto\inf\,\left\{\sum_{k\in\mathbb N}\,\lambda_f([a_k,b_k)):A\subset\bigcup_{k\in\mathbb N}[a_k,b_k)\right\}, \end{align} $
which can be shown to be an outer measure. (Therefore, with $\theta^*_f$, Carathéodory's method generates the measure $\mu_f$, the Lebesgue-Stieltjes measure.)
What changes in this rationale if we no longer assume $f$ has to be continuous?