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Let $f: \mathbb{R} \rightarrow \mathbb{R}$, and for $x \in \mathbb{R}$, let $T_f(x)=\sup \left\{\sum_{j=1}^N \left|f(x_j)-f(x_{j-1})\right|\right\}$, where the supremum is taken over all $N \in \mathbb{N}$ and all $(x_j)$ such that $-\infty<x_0<x_1<\ldots < x_N=x$.

We say that a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is of bounded variation iff the function $T_f$ is bounded.

We say that a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is in NBV iff $f$ is of bounded variation left-continuous and $f(x)\rightarrow 0$ if $x \rightarrow -\infty$.

In Rudin, Real and complex analysis, Thr.8.14 states that if $\mu$ is finite real Borel measure then the function $f(x)=\mu((-\infty,x))$ for $x \in \mathbb{R}$ is in NBV and that conversely, for $f\in NBV$ there exists exactly one real finite Borel measure $\mu$ such that $f(x)=\mu((-\infty,x))$ for $x \in \mathbb{R}$. Moreover, $T_f(x)=\|\mu\|((-\infty,x))$ for $x \in \mathbb{R}$.

My question is : is there a Jordan decomposition of $f \in NBV$ (it is very known when $f$ is defined on $[a,b]$) and is there a connection between the Jordan decomposition of real finite Borel measure $\mu$ for which $f$ is "distribution function" (i.e. $f(x)=\mu((-\infty,x))$ for $x \in \mathbb{R}$) and a Jordan decomposition of $f$?

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