The famous ( ? ) Jordan decomposition Theorem for Bounded variation functions is generally stated as below.
Theorem. If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+(b)-f^+ (a) + f^- (b)- f^- (a)$ where $TV (f)$ denotes the total variation of $f$. The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.
My question is motived by Hahn-Jordan decomposition Theorem to signed measure.
Question 1: It is possible to generalize the theorem of Jordan decomposition for functions of bounded variation defined on intervals [a, b] for functions of bounded variation defined on some mensurable space $(\Omega,\mathcal{F})$ with some partial order $\prec$?
In this case, a function $f$ is nondecreasing if $\omega \prec \eta$ implies $f(\omega)\leq f(\eta)$.
Of course, to generalize this theorem we must define the that becomes the total variation VT $ (f) $ for functions $ f $ defined in $ \Omega $.
In this case I prefer to be flexible and not risk any definition not to restrict the generality of the solution.
Question 2:Suppose that the theorem admits no such generazation. Now if $(\Omega,d) $ is a compact metric space with a partial order $\prec$ and the functions $ f $ are continuous then there is some generalization for this case?
Thank you.