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Strongly measurable function

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Strong measurability has a number of different meanings, some of which are explained below.

Values in Banach spaces

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For a functionf with values in aBanach space (orFréchet space),strong measurability usually meansBochner measurability.

However, if the values off lie in the space ${\mathcal {L}}(X,Y)$ ofcontinuous linear operators fromX toY, then oftenstrong measurability means that the operatorf(x) is Bochner measurable for each fixedx in the domain off, whereas the Bochner measurability off is calleduniform measurability (cf. "uniformly continuous" vs. "strongly continuous").

Bounded operators

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A family of bounded linear operators combined with thedirect integral is strongly measurable, when each of the individual operators is strongly measurable.

Semigroups

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Asemigroup 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

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

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

v t e Analysis intopological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector-valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner /Weakly /Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

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