Measurable Functions on Hilbert Spaces

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

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

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.