Strongly measurable functions

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), strong measurability usually means Bochner measurability.

However, if the values of f lie in the space $\mathcal{L}(X,Y)$ of continuous linear operators from X to Y, then often strong measurability means that the operator f(x) is Bochner measurable for each $x\in X$, 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.^[1] It is uniformly measurable if and only if it is uniformly continuous, i.e., if and only if its generator is bounded.

References

1. Example 6.1.10 in Linear Operators and Their Spectra, Cambridge University Press (2007) by E.B.Davies