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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 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").

Bounded operators

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.


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