Prescription of measures on functional spaces by means of numerical densities and path integrals
Let us consider a separable real Hilbert space \(H\). We choose an orthonormal basis \(h_ 1\), \(h_ 2,\dots,\) in it, denote the linear hull of the vectors \(h_ 1,\dots,h_ n\) by \(H_ n\), and set \(H_ 0=\bigcup^ \infty_{n=1} H_ n\). We suppose that a bounded positive function \(F\), whose restriction to each space \(H_ n\) is measurable, is defined on \(H_ 0\). For each \(p \in H_ 0\) we compute \[ \chi(p)=c_ n^{-1} \int_{\mathbb{R}^ n} e^{i \sum^ n_{k=1} x_ k (p,h_ k)} F \left( \sum^ n_{k=1} x_ kh_ k \right) dx_ 1\dots dx_ n, \] where \[ c_ n=\int_{\mathbb{R}_ n} F \left( \sum^ n_{k=1} x_ kh_ k \right) dx_ 1 \dots dx_ n. \] We assume that there exists an increasing sequence of natural numbers \(n_ 1\), \(n_ 2,\dots,\) such that \(\exists \chi (p)=\lim \chi_{n_ k} (p)\) exists \(\forall p \in H_ 0\) and a probability measure \(\mu\) with the characteristic function \(\chi\) exists on a certain completion of the space \(H_ 0\). We call the function \(F\) the density of the measure \(\mu\). We formulate sufficient conditions under which a function \(F\) is the density of a certain measure \(\mu\) and describe properties of this measure. We also discuss the problem about the class of the measures that have a density and the uniqueness problem.
scientific article; zbMATH DE number 589623