One-node Gauss data are midpoint data

Given: Lebesgue measure on $[0,1]$ and the one-node Gaussian quadrature rule from The m-node Gaussian quadrature rule for a positive measure.

Lebesgue measure on the unit interval supplies Gaussian moment data: Lebesgue measure on $[0,1]$ is valid one-node moment data through degree two, and its support has at least two points.

Positive-measure Gaussian quadrature moment data: the order-two datum also meets the order-one branch used by the rule definition, because its measure, support, and moments through degree one satisfy that branch's conditions.

Monic orthogonal polynomials exist uniquely through the quadrature order: for these data, the monic degree-one polynomial orthogonal to constants is unique.

The m-node Gaussian quadrature rule for a positive measure and The Gaussian nodes, cardinal polynomials and weights are well-defined: the Gaussian node is the root of that polynomial and its weight is the integral of the one-node cardinal polynomial; the root and weight are defined.

Cardinal-polynomial interpolation: the cardinal polynomial for one node is ${\mathrm{\ell }}_1(x)=1$.

Riemann and Lebesgue integrals agree for continuous data on [0,1] and The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: for continuous $1$ and $x$ on $[0,1]$, Lebesgue and Riemann integrals agree, and the Riemann integrals are computed by primitives.

Collocation tableau from distinct quadrature nodes: the one-stage collocation coefficient is $Q_{11}={\int }_0^{c_1}{\mathrm{\ell }}_1(s)ds$ and the endpoint weight is $b_1={\int }_0^1{\mathrm{\ell }}_1(s)ds$.

Countable Choice, through the Lebesgue interval-measure theorem used by [F1].