The Theory of Stochastic Processes I pp 525-557 | Cite as

# Measurable Functions on Hilbert Spaces

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## Abstract

Consider a measurable Hilbert space (ℋ,𝓑) on which a measure that the domain of definition

*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$$ \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}\left( {\alpha x + \beta y} \right) = \alpha \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(x) + \beta \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(y) $$

*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.## Keywords

Hilbert Space Measurable Function Orthogonal Polynomial Homogeneous Polynomial Linear Manifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2004