Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps. Then $J$ is a vector space and contains the following subsets: all continuous functions $C$, all (r.c.l.l.) non-decreasing functions $M$ and thus distribution functions $D \subseteq M$ of probability measures on $[0, \infty)$, the linear hull $M - M$ of (r.c.l.l.) functions of locally bounded variation and many others.
For $f_n, f \in D$ define the convergence $f_n \to f :\Leftrightarrow f_n(x) \to f(x)$ for all continuity points $x$ of $f$. Then $\to$ satisfies:
- if $f_n \to f$ and $f_n \to g$ then $f = g$ (by right-continuity)
- if $f_n = f$ for all $f$ then $f_n \to f$
- if $f_n \to f$ then $f_{n_k} \to f$ for all subsequences $f_{n_k}$ of $f_n$
It follows that $\to$ is an $L$-convergence on $J$ and defines a sequential $T_1$ topology $\tau$ on $J$. Clearly, $\tau$ is weaker than the topology of (everywhere) pointwise convergence. Note also that the usual notion of "uniform convergence on a particular family $\mathcal{F}$ of subsets of $[0,\infty)$" (e.g. pointwise convergence, uniform convergence or compact convergence) is based on a specification of $\mathcal{F}$ in advance whereas for the convergence $\to$ we choose such a family depending on properties of its limit.
I have only met this notion of convergence on the subset $D$ of distribution functions in which case it is just the weak topology of probability measures. I haven't seen discussions of properties of the topologies induced by this notion of convergence on larger spaces like $J$, but it seems to "naturally" extend the convergence on $D$.
Questions:
- Does $\to$ has a special name or does anyone know for references regarding properties of this convergence / topology (e.g. descriptive properties or compatibility with vector space structure)?
- Similarly, one can also consider the (locally) uniform convergence at points of continuity of the limit function. Is there also information available on this convergence?
