I was reading Terence Tao's notes on Analysis. He says Theorem 13(g) cannot be extended from Riemann integral to Riemann-Stieltjes integral:
Most (but not all) of the remaining theory from Week 9 notes then can be carried over without difficulty, replacing Riemann integrals with Riemann-Stieltjes integrals and lengths with $\alpha$-length. (There are a couple results which break down; Theorem 13(g), Proposition 16, and Proposition 17 are not necessarily true when $\alpha$ is discontinuous at key places (e.g. if $f$ and $\alpha$ are both discontinuous at the same point, then $\int_{I}fd\alpha$ is unlikely to be defined).
Source: page 4 of week 10 notes
But I do not see why Theorem 13(g) breaks down if we extend it to Riemann-Stieltjes integral. Can anyone explain? Below is the theorem:
Theorem 13 (Laws of integration). Let $I$ be a generalized interval, and let $f: I \rightarrow R$ and $g: I \rightarrow R$ be Riemann integrable functions on I.
...
(g) Let $J$ be a generalized interval containing $I$ (i.e. $I \subseteq J$), and let $F: J \rightarrow R$ be the function
$$ F(x) := \left\{ \begin{array}{ll} f(x) & \text{if } x \in I \\ 0 & \text{if } x \notin I \end{array} \right. $$
Then $F$ is Riemann integrable on $J$, and $\int_{J}F = \int_{I}f$.
Source: page 14 of week 9 notes