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

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

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