Lebesgue measure on the unit interval supplies Gaussian moment data

Given: An integer $M\ge 1$ and Lebesgue measure $\lambda$ on $[0,1]$.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: $\lambda ([0,1])=1$, and every nonempty relative interval neighborhood in $[0,1]$ has positive measure equal to its length.

Integrable real and complex functions, and their integrals and Monotonicity and nonnegative homogeneity of the nonnegative integral: if $g$ is Borel measurable and $0\le g\le 1$ on $[0,1]$, then $g$ is integrable and $0\le \int gd\lambda \le 1$ because $\lambda ([0,1])=1$.

Positive-measure Gaussian quadrature moment data: order-$2M$ data require a finite positive measure whose relative support contains at least $M+1$ points and whose absolute moments through degree $2M$ are finite.

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function and The Borel sigma-algebra of a topological space: polynomial and absolute-value functions are continuous on $[0,1]$. Continuity makes preimages of open subsets of $\mathbb{R}$ relatively open in $[0,1]$, so preimages of Borel sets belong to the relative Borel sigma-algebra generated by those opens.

Countable Choice, required by [F1] for the Lebesgue interval-measure theorem.