Weak type (1,1) ($\mu \{Sf>\lambda \}\le C\Vert f{\Vert }_1/\lambda$)

For a nonnegative sublinear operator, weak type $(1,1)$ is the displayed level-set bound for all integrable inputs and positive thresholds. It supplies an exceptional-set estimate for approximation errors; the uncentered maximal weak-type inequality is a central example.

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.