My question has to do with continuity requirements for functions used to define a Lebesge-Stieltjes measure on $\mathbb{R}$. In Fremlin's first volume of Measure Theory, the Lebesgue measure on $\mathbb{R}$ is defined by considering half-open intervals of the form $[a,b)=\{x:a\leq x<b\}$. The length $\lambda I$ of a half-open interval $I$ is defined by setting $$\lambda\varnothing = 0,\quad \lambda[a,b)=b-a\text{ if }a<b.$$ From there, the Lebesgue outer-measure $\theta:2^\mathbb{R}\to[0,\infty]$ is defined in the usual way by setting \begin{align*} \theta A= \inf\bigg\{\sum_{j=0}^\infty \lambda I_j:\langle I_j\rangle_{j\in\mathbb{N}}\text{ is a sequence of half-open intervals} \\ \text{such that } A\subseteq\bigcup_{j\in\mathbb{N}} I_j\bigg\} \end{align*} for any $A\subseteq \mathbb{R}$. (Above, half-open specifically means open on the right and closed on the left.) One of the exercises in the book replaces $\lambda$ in the above definition with $\lambda_g$, where $g\colon\mathbb{R}\to\mathbb{R}$ is an arbitrary non-decreasing function and $$\lambda_g\varnothing = 0,\quad\lambda_g[a,b)=\lim_{x\to b^-}g(x)-\lim_{x\to a^-} g(x)\text{ if }a<b.$$ The corresponding outer measure is denoted $\theta_g$ in the textbook. A subsequent question asks what happens if we were to naively define $\lambda_g[a,b)=g(b)-g(a)$.
I understand that in the first case, we still have a legitimate outer measure whose collection of measurable sets contains all Borel sets. Is it true that the same holds for the second case as well? I couldn't find much that would change, except in the second case, we would no longer have $\lambda_g I=\theta_g I$ for any half-open interval $I$.
If this is true, and the naive definition still defines a legitimate outer measure on $\mathbb{R}$, is there any context in which the naive definition is worth studying (even if $g$ is not left-hand continuous)?
Your are right the only place where there is a breakdown with the second definition is 114Bb where you need the property $g(\sup_{x\in A}x)=\sup_{x\in A} g(x)$, which in general only holds if $g$ is non-decreasing and left-continuous.
I don't know any context where it would be advantageous to use a non-decreasing function $g$ which is not left-continuous. In any case you can always make it left-continuous by a simple redefinition.