Strongly measurable function

Strong measurability has a number of different meanings, some of which are explained below.

Values in Banach spaces

For a function f with values in a Banach space (or Fréchet space) X, strong measurability usually means Bochner measurability.

However, if the values of f lie in the space ${\mathcal {L}}(X,Y)$ of continuous linear functionals from X to Y, then often strong measurability means that fx is Bochner measurable for each $x\in X$, whereas the Bochner measurability of f is called uniform measurability (cf. "uniformly continuous" vs. "strongly continuous").

Semi-groups

A semigroup of linear operators can be strongly measurable yet not strongly continuous. It is uniformly measurable if and only if it is uniformly continuous, i.e., if and only if its generator is bounded.

Strong measure zero sets

A set of reals is said to have strong measure zero if for every infinite sequence of positive numbers there is a corresponding sequence of intervals with the given lengths whose union covers the set. Borel's conjecture that all such sets are countable, is undecidable in ZFC.