= Almost-everywhere convergence from weak-type domination
Suppose a family of <linear operators>, including its limiting operator, is pointwise dominated by a <sublinear operator> of <weak type (1,1)>. Convergence <almost everywhere> on an $L^1$-dense subspace then extends to every $L^1$ input. Approximate the input by errors of <norm> at most $2^{-2n}$. The weak estimate and the <Borel-Cantelli first lemma> make the corresponding domination errors tend to zero <almost everywhere>. <Linearity> bounds the difference between the full and approximating convergence errors by twice that domination error. This argument does not need an uncountable supremum to be measurable.
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