Generalizing a special case of Lebesgue decomposition for monotone functions

Question

From Lemma $62$ of Math $245A$ note $5$:

Continuous-singular decomposition for monotone functions: Let ${F: {\bf R} \rightarrow {\bf R}}$ be a bounded, monotone non-decreasing function. Then ${F}$ can be expressed as the sum of a continuous monotone non-decreasing function ${F_c}$ and a jump function ${F_{pp}}$.

It can be shown that the decomposition is given by setting $F_{pp} := \sum_{x \in A} c_xJ_x$, where the jump $${c_x := F_+(x) - F_-(x) > 0}$$ and the fraction for the basic jump function $J_x$ $${\theta_x := \frac{F(x)-F_-(x)}{F_+(x)-F_-(x)} \in [0,1]}$$

are defined for points $x$ in $A$ - the set of discontinuities of $F$; and then setting $F_c := F - F_{pp}$.

Problem: Find a suitable generalisation of the notion of a jump function that allows one to extend the above decomposition to unbounded monotone functions, and then prove this extension. (Hint: the notion to shoot for here is that of a “locally jump function”.)

Attempt: The restriction of any monotone function to a compact interval $[a,b]$ is bounded, so if one extends $F \downharpoonright_{[a,b]}$ to ${\bf R}$ by setting $F(x) := F(a)$ for $x < a$ and $F(x) := F(b)$ for $x > b$, then $F$ is bounded, and the Lemma applies. As such, we define a locally jump function to be a function that agrees with a jump function on any compact interval, and claim that $F$ can be expressed as the sum of a continuous monotone non-decreasing function $F_c$ and a locally jump function $F_{pp}$. For any interval $[n,n+1]$, denote the above extension by $F_n$, with \begin{align*} F_n := F_{c}^n + F_{pp}^n \end{align*} as given by the Lemma. I was trying to show that $$F_c := \sum_{n \in {\bf Z}} F_c^n \downharpoonright_{(n,n+1]},\ F_{pp} := \sum_{n \in {\bf Z}} F_{pp}^n \downharpoonright_{(n,n+1]}$$ is the desired decomposition, but ran into issues showing that the continuous $F_c^n$ agree at the endpoints. Namely, that $F_c^n(n+1) = F_c^{n+1}(n+1)$ for all $n$. Is this argument on the right track?

For context, the definition of a jump function is given here: Uniqueness of a special case of Lebesgue decomposition.

Edit: With the generalization of the notion of a jump function to locally jump function, it seems that no further changes on the original decomposition are needed.

Answer 1

Let $F: {\bf R} \rightarrow {\bf R}$ be a monotone non-decreasing function, and $A$ the set of discontinuities of $F$, which is at most countable. We define a locally jump function to be a function that is a jump function on any compact interval $[a,b]$, and claim that $F$ can be expressed as the sum of a continuous monotone non-decreasing function $F_c$ and a locally jump function $F_{pp}$.

For each $x \in A$, we define the jump $c_x := F_+(x) - F_-(x) > 0$, and the fraction $\theta_x := \frac{F(x)-F_-(x)}{F_+(x)-F_-(x)} \in [0,1]$. Thus \begin{align*} F_+(x) = F_-(x) + c_x \hbox{ and } F(x) = F_-(x) + \theta_x c_x. \end{align*} Note that $c_x$ is the measure of the interval $(F_-(x),F_+(x))$. By monotonicity, these intervals are disjoint. Since $F$ is bounded on $[a,b]$, the union of these intervals for $x \in A \cap [a,b]$ is bounded. By countable additivity, we thus have $\sum_{x \in A \cap [a,b]} c_x < \infty$, and so if we let $J_x$ be the basic jump function with point of discontinuity $x$ and fraction $\theta_x$, then the function \begin{align*} F_{pp} := \sum_{x \in A} c_xJ_x \end{align*} is a locally jump function. $F$ is discontinuous only at $A$, and for each $x \in A$ one easily checks that \begin{align*} (F_{pp})_+(x) = (F_{pp})_-(x) + c_x \hbox{ and } F_{pp}(x) = (F_{pp})_-(x) + \theta_x c_x \end{align*} where $(F_{pp})_-(x) := \lim_{y \rightarrow x^-} F_{pp}(y)$, and $(F_{pp})_+(x) := \lim_{y\rightarrow x^+} F_{pp}(y)$. We thus see that the difference $F_c := F-F_{pp}$ is continuous. Finally, we verify that $F_c$ is monotone non-decreasing. By continuity it suffices to verify this away from the (countably many) jump discontinuities, thus we need \begin{align*} F_{pp}(b)-F_{pp}(a) \leq F(b)-F(a) \end{align*} for all $a < b$ that are not jump discontinuities. But the LHS can be rewritten as $\sum_{x \in A \cap [a,b]} c_x$, while the RHS is $F_-(b) - F_+(a)$. As each $c_x$ is the measure of the interval $(F_-(x), F_+(x))$, and these intervals for $x \in A \cap [a,b]$ are disjoint and lie in $(F_+(a),F_-(b))$, the claim follows from countable additivity.