Abstract
Consider a measurable Hilbert space (ℋ,𝓑) on which a measure μ is defined. Every continuous linear functional l(x) defined on ℋ is clearly 𝓑-measurable. It is known that if a sequence of continuous linear functionals l n (x) converges to a certain limit l(x) for all x then this limit will also be a continuous linear functional on ℋ. The situation, however becomes different if we require that l n (x) possess the limit not for all x but only on a set D such that μ(D)= 1. It is natural to refer to such limiting functions l(x) as 𝓑-measurable functionals. These functions, being limits of sequences of measurable functions will also be 𝓑-measurable. It follows from the relations
that the domain of definition D l of functional l(x) is a linear manifold and that l(x) is a linear (additive and homogeneous) functional. (We assume that the functional l(x) is defined wherever the corresponding limit exists.) Hereafter we shall consider non-degenerate measures μ such that μ(L) = 0 for any proper subspace L of the space ℋ.
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© 2004 Springer-Verlag Berlin Heidelberg
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Gihman, I.I., Skorokhod, A.V. (2004). Measurable Functions on Hilbert Spaces. In: The Theory of Stochastic Processes I. Classics in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61943-4_8
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DOI: https://doi.org/10.1007/978-3-642-61943-4_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20284-4
Online ISBN: 978-3-642-61943-4
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