Abstract
This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.
Highlights
Over the years, mathematics scholars have studied inner product functions in Banach spaces
The study proves the existence of integral representation of pointwise product vector measure duality with values in a Hilbert space
If ψ(A×B) is a vector valued function, where A × B ∈ ρ × ε is a fixed set, the product of ψ(A×B) and Ttμik ×νik is given by ψ(A×B) ∗ Ttμik ×νik = Ttμik ×νik (A × B) ∈ X × Y If b ∈ B is a fixed element, the set (A × B)b is measurable with respect to ρ
Summary
Mathematics scholars have studied inner product functions in Banach spaces. Many theories on integration of vector valued functions with respect to vector measure duality have been proved. The study proves the existence of integral representation of pointwise product vector measure duality with values in a Hilbert space. The functions (μi)i∈I and (νi)i∈I defined on sigma rings ρ and ε with values in Hilbert spaces X and Y respectively i.e μi : ρ → X and νi : ε → Y for each i ∈ I, are called vector measures. There exists a vector measure function git for t ∈ R in M (ρ × ε, X × Y ), where M (ρ × ε, X × Y ) is a set of X × Y valued vector measures defined on ρ × ε
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