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.

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References

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

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

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory