This is a question regarding a statement on page 7 of the lecture notes https://vittoriasilvestri.files.wordpress.com/2021/11/lecture_notes_2017.pdf.
Let $a:[0,\infty)\to\mathbb{R}$ be cadlag (right-continuous with left limits) and non-decreasing, with $a(0)=0$. Then there exists a unique Borel measure $da$ on $[0,\infty)$ such that for all $s<t$ we have $$da((s,t]) = a(t)-a(s).$$
Then, for any measurable and integrable $f:\mathbb{R}\to\mathbb{R}$, we can define the (Lebesgue-Stieltjes) integral $$(f\cdot a)(t)=\int_{(0,t]}f(s)da(s).$$
The author then mentions that if $a$ is continuous then $f\cdot a$ is continous and we can unambiguously write $$\int_{(0,t]}f(s)da(s) = \int_0^tf(s)da(s).$$
I'm relatively new to measure-theoretic integrals and so I am unsure as to how the integral on the RHS above has been defined, and why it requires continuity for the identity to be true. Additionally, is the RHS integral defined in the case of non-continuity of the LHS, and it is just that the equality does not hold? Any advice would be greatly appreciated!