= Dense approximation under a weak-type maximal bound
Suppose <linear operators> $A_t$ and $A_0$ on a <normed space> satisfy $|A_tf|,|A_0f|\leq Sf$, where $S$ has <weak type with a normed domain>, and all relevant suprema are measurable. If $A_tg\to A_0g$ <almost everywhere> on a dense subspace, the same holds for every input. For a fixed approximation $g$, the limiting error is at most $2S(f-g)$. The measure where it exceeds $a$ is at most $(2C\|f-g\|/a)^q$, which tends to zero as the approximation improves. This is a convergence-to-a-specified-limit version of <almost-everywhere convergence from weak-type domination>.
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