Strongly measurable function

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

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

A family of bounded linear operators combined with the direct integral is strongly measurable, when each of the individual operators is strongly measurable.

A semigroup of linear operators can be strongly measurable yet not strongly continuous.^[1] It is uniformly measurable if and only if it is uniformly continuous, i.e., if and only if its generator is bounded.