Almost basic convergence is a mode of convergence for sequences of functions (or signed measures via their distribution functions) introduced by Stanek (2024) as a relaxation of basic convergence.
Let (f_n) and f be functions from R to R. We say (f_n) converges almost basically to f, written f_n =alm⇒ f, if every subsequence (f_{n_k}) contains a further subsequence (f_{n_{k_l}}) such that
f_{n_{k_l}}(x) - f_{n_{k_l}}(y) → f(x) - f(y)
for all x, y in R \ S, where S is a set of Lebesgue measure zero (rather than merely countable as in basic convergence).
The distinction from basic convergence is subtle but real: basic convergence requires the exceptional set S to be at most countable, while almost basic convergence allows S to be any Lebesgue-null set. Stanek constructs a sequence using Smith-Volterra-Cantor sets (Lemma 3.2) showing that almost basic convergence is strictly weaker than basic convergence.
Almost basic convergence has better topological properties than basic convergence: it is metrizable on the space W of right-continuous functions of bounded variation vanishing at -infinity. The metric is a Ky Fan-type metric (Lemma 3.4):
d(f, g) = min{eps >= 0 : lambda({x in [-1/eps, 1/eps] : |f(x) - c - g(x)| > eps}) ⇐ eps for some c in R}
However, the metric space (W, d) is separable but NOT complete (Lemma 3.6), and the convergence is not induced by any norm (Lemma 3.5).
In the main characterisation (Theorem 3.12), almost basic convergence can replace basic convergence: vague convergence ⇐> almost basic convergence + local uniform boundedness in variation.
Key Details
- Relaxation of: basic convergence — allows Lebesgue-null exceptional sets instead of countable
- Strictly weaker: there exist sequences converging almost basically but not basically
- Metrizable: via a Ky Fan-type metric, unlike basic convergence
- Not complete: the metric space is separable but incomplete
- Main equivalence: vague convergence ⇐> almost basic convergence + local uniform boundedness in variation
- For non-negative measures: equivalent to vague convergence and basic convergence (hierarchy collapses)