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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
(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 In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
* the final topology on the disjoint union
* the top ...
").
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.
Semigroups
A
semigroup of linear operators can be strongly measurable yet not strongly continuous.
[ Example 6.1.10 in Linear Operators and Their Spectra, Cambridge University Press (2007) by E.B.Davies] It is uniformly measurable if and only if it is uniformly continuous, i.e., if and only if its generator is bounded.
References
{{algebra-stub
Banach spaces
Semigroup theory