Positive-measure Gaussian quadrature moment data

Let $a<b$ be real numbers, let $m\ge 1$ be an integer, and let $r\in \{2m-1,2m\}$. A Gaussian quadrature moment datum of order $r$ is a finite positive measure $\mu$ on the relative Borel sigma-algebra $\mathcal{B}([a,b])$ (The Borel sigma-algebra of a topological space, Measure spaces, Measures on sigma-algebras, Finite, sigma-finite, and semifinite measures) such that $\mu ([a,b])>0$, and whose relative support $S_{\mu }:=\{x\in [a,b]:\mu (U)>0\text{ for every relatively open }U\subseteq [a,b]\text{ with }x\in U\}$ contains at least $m+1$ distinct points. For every integer $0\le k\le r$, require $x\mapsto \mathrm{\mid }x{\mathrm{\mid }}^k$ to be integrable with respect to $\mu$ (Integrable real and complex functions, and their integrals) and put $M_k:={\int }_{[a,b]}{x}^{k}d\mu (x)$. Powers use the natural-exponent recursion of Integer powers $a^m$, with $x^0=1$ for every real $x$, including $x=0$; thus $M_0$ is the integral of the constant function $1$.

The branch $r=2m-1$ supplies all moments used to construct the monic degree-$m$ orthogonal polynomial and prove rule exactness. The stronger branch $r=2m$ also defines its squared norm and is used for the remainder formula. Because $[a,b]$ is bounded and $\mu$ is finite, these absolute-moment integrability conditions follow from finiteness; they are stated to make the two moment interfaces explicit.