 Dense approximation under a weak-type maximal bound

Suppose linear operators $A_t$ and $A_0$ on a normed space satisfy $|A_{t}f|,|A_{0}f|\le Sf$, where $S$ has weak type with a normed domain, and all relevant suprema are measurable. If $A_{t}g\to A_{0}g$ 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 $(2C\Vert f-g\Vert /a{)}^q$, 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.