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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.

Use the approximants just chosen. The weak type (1,1) bound givesThe measures of these exceptional sets are summable. Indeed, the measure of their union for $n\ge m$ is at most $C{\sum }_{n\ge m}{2}^{-n}\to 0$. Hence their limsup is null; this is the Borel-Cantelli first lemma. Outside it, $S(g-f_n)\to 0$.

Remove also the countable null set from part (b). At each remaining point and for every $n$, the estimate in part (a) yieldsLetting $n\to \infty$ makes the right side zero. ThereforeThis proves almost-everywhere convergence from weak-type domination. The argument uses countably many measurable exceptional sets and does not assume that a supremum over all real parameters is measurable.