I am trying to prove that the following three assertions about a function $f$ on an interval $[a, b]$ are equivalent, but I am encountering some difficulties:
$f$ is the difference of two increasing right-continuous functions.
There exists a signed measure $\mu$ on $[a, b]$ such that $f(x)=\mu([a, x])$ for everey $x \in[a, b]$.
$f$ is right-continuous and of bounded variation in the sense that
$$ \sup _{a=a_{0}<a_{1}<\ldots<a_{n-1}<a_{n}=b} \sum_{i=1}^{n}\left|f\left(a_{i}\right)-f\left(a_{i-1}\right)\right|<\infty $$
where the supremum is over all choices of the integer $n \geq 1$ and the reals $a_{0}, \ldots, a_{n}$ such that $a=a_{0}<a_{1}<\ldots<a_{n-1}<a_{n}=b$.
I started by showing that assertion 1 implies assertion 3 using the definition of bounded variation and the properties of increasing functions. Since increasing functions are of bounded variation, their difference is also of bounded variation, which covers the supremum condition in assertion 3.
Transitioning from assertion 1 to 2, I used the fact that any increasing function can be associated with a measure (for instance, Lebesgue-Stieltjes measure), and thus their difference would give rise to a signed measure, satisfying assertion 2.
However, I am stuck trying to prove that assertion 2 implies assertion 1 and that assertion 3 implies the others. For assertion 2 to 1, I thought to represent $\mu$ as the difference of two measures corresponding to the positive and negative parts of $f$, but I'm unsure how to formalize this or if it's the right approach.
As for assertion 3 implying the others, I understand that functions of bounded variation can be decomposed into increasing functions, but I'm not sure how to incorporate right-continuity into this decomposition or how to construct the signed measure for assertion 2.
Could someone provide some guidance or references that could help me formalize these proofs? Any help in bridging the gaps in my solution attempt would be greatly appreciated.