This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen elsewhere.
Transfinite summability of series.
Let $(S,\leq)$ be a totally ordered space. In this post,
- an initial segment of $(S,\leq)$ is a set $I \subset S$ such that for every $\alpha \in I$, we have $\{\beta \in S : \beta \leq \alpha\} \subset I$;
- an interval of $(S,\leq)$ is a set $I \subset S$ such that for every $\alpha,\beta \in I$ with $\alpha \leq \beta$, we have $\{\gamma \in S : \alpha \leq \gamma \leq \beta \} \subset I$.
Also, for any partition $\mathcal{P}$ of $S$ and any $J \subset S$, we write $\mathcal{P}(J) \subset \mathcal{P}$ for the set of elements of $\mathcal{P}$ that contain at least one element of $J$.
Given a pairwise-disjoint collection $\mathcal{C}$ of non-empty intervals of $(S,\leq)$, we write $\leq_\mathcal{C}$ for the total ordering of $\mathcal{C}$ that is naturally inherited from $\leq$.
A nice partition of $(S,\leq)$ is a partition $\mathcal{P}$ of $S$ such that every element of $\mathcal{P}$ is a non-empty interval of $(S,\leq)$ and the total ordering $\leq_\mathcal{P}$ is a well-ordering.
Now let $V$ be a Banach space (or just a complete normed abelian group). For any $\mathbf{x}=(x_\alpha)_{\alpha \in S} \in V^S$, an $\mathbf{x}$-nice partition of $(S,\leq)$ is a nice partition $\mathcal{P}$ of $(S,\leq)$ such that for each $I \in \mathcal{P}$, $$ \sum_{\alpha \in I} \| x_\alpha \| < \infty. $$ Given an $\mathbf{x}$-nice partition $\mathcal{P}$ of $(S,\leq)$, we define for each initial segment $J$ of $(S,\leq)$ a value $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha \, \in \, V \cup \{\text{NaN}\} $$ by the following recursive procedure:
- In the case that $J=\emptyset$: Define $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha = 0. $$
- In the case that $\mathcal{P}(J)$ has a maximum element $I$ with respect to $\leq_\mathcal{P}$: Define $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha = \left( \sum_{\alpha \in \bigcup(\mathcal{P}(J) \setminus \{I\})}^{\mathcal{P},\leq} x_\alpha \right) + \sum_{\alpha \in I \cap J} x_\alpha $$ if $$ \sum_{\alpha \in \bigcup(\mathcal{P}(J) \setminus \{I\})}^{\mathcal{P},\leq} x_\alpha \in V; $$ otherwise, define $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha = \text{NaN}. $$
- In the case that $J \neq \emptyset$ and $\mathcal{P}(J)$ has no maximum element with respect to $\leq_\mathcal{P}$ (and hence $J=\bigcup(\mathcal{P}(J))$): Suppose there exists $x \in V$ with the property that for every neighbourhood $U \subset V$ of $x$, there exists $\gamma_U \in J$ such that for every $\beta \in J$ with $\gamma_U \leq \beta$, $$ \sum_{\alpha \leq \beta}^{\mathcal{P},\leq} x_\alpha \in U; $$ then we define $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha = x. $$ But if no such $x \in V$ exists, then we define $$ \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha = \text{NaN}. $$
Proposition. Given two $\mathbf{x}$-nice partitions $\mathcal{P}_1$ and $\mathcal{P}_2$ of $(S,\leq)$, we have that for every initial segment $J$ of $(S,\leq)$, $$ \sum_{\alpha \in J}^{\mathcal{P}_1,\leq} x_\alpha = \sum_{\alpha \in J}^{\mathcal{P}_2,\leq} x_\alpha. $$
The proof is a straightforward transfinite induction on the initial segments of $(\mathcal{P},\leq_\mathcal{P})$ with $\mathcal{P}=\{I_1 \cap I_2 : I_1 \in \mathcal{P}_1,\, I_2 \in \mathcal{P}_2\} \setminus \{\emptyset\}$.
So then, given $\mathbf{x} \in V^S$, if there exists an $\mathbf{x}$-nice partition of $(S,\leq)$, then for each initial segment $J \subset S$ we can simply define $$ \sum_{\alpha \in J}^\leq x_\alpha $$ without specifying the $\mathbf{x}$-nice partition $\mathcal{P}$. Note that, by construction, for initial segments $J_1,J_2 \subset S$ with $J_1 \subset J_2$, we have $$ \sum_{\alpha \in J_2}^\leq x_\alpha = \sum_{\alpha \in J_2}^{\leq|_{J_1}} x_\alpha $$ where $\leq|_{J_1}$ denotes the inherited ordering on $J_1$ from $(S,\leq)$.
Definition. We say that $\mathbf{x} \in V^S$ is summable if there exists an $\mathbf{x}$-nice partition of $(S,\leq)$ and $$ \sum_{\alpha \in S}^\leq x_\alpha \neq \text{NaN}. $$
Note that, by construction, if $\mathbf{x}$ is summable then every initial segment $J \subset S$ has $$ \sum_{\alpha \in J}^\leq x_\alpha \neq \text{NaN}. $$
From now on, we will omit the $\leq$ above the $\sum$ sign.
My questions.
Given a càdlàg function $f \colon [0,1] \to \mathbb{R}$, we write $\mathcal{D}(f) \subset (0,1]$ for the set of discontinuity points of $f$.
- Has the above notion of summability been studied before? Does it have a name?
- Given a càdlàg function $f \colon [0,1] \to \mathbb{R}$, is $(f(t)-f(t-))_{t \in \mathcal{D}(f)}$ necessarily summable?
- Given a càdlàg function $f \colon [0,1] \to \mathbb{R}$ for which $(f(t)-f(t-))_{t \in \mathcal{D}(f)}$ is summable, do we necessarily have that the function $\tilde{f} \colon [0,1] \to \mathbb{R}$ given by $$ \tilde{f}(t) = f(t) - \sum_{s \in \mathcal{D}(f) \cap [0,t]} (f(s)-f(s-)) $$ is continuous?
Update. Anthony Quas has provided a negative answer to my second question (and hence the third question is arguably not very interesting).
But forgetting about jumps of càdlàg functions, I'm still quite interested in my first question: Has my above notion of summability been studied before? It feels like a very natural generalisation of $\sum_{n=1}^\infty x_n$ for non-$l^1$ sequences $(x_n)_{n \in \mathbb{N}}$ to the more general case of $S$-indexed families of numbers $(x_\alpha)_{\alpha \in S}$ for totally ordered sets $S$ besides $\mathbb{N}$.
