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 [math]\displaystyle{ \mathcal{L}(X,Y) }[/math] of continuous linear operators from X to Y, then often strong measurability means that the operator f(x) is Bochner measurable for each [math]\displaystyle{ x\in X }[/math], whereas the Bochner measurability of f is called uniform measurability (cf. "uniformly continuous" vs. "strongly continuous").[clarification needed]
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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