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The Jacobi weight defines a finite positive measure with full support

Statement

Assume Countable Choice. Let $α, β > - 1$ and define $w_{α, β} (t) = {\begin{cases} (1 - t)^{α} (1 + t)^{β}, & - 1 < t < 1, \\ 0, & t = \pm 1. \end{cases}$ Let $λ$ be Lebesgue measure restricted to the Borel subsets of $[- 1, 1]$ , and set $μ_{α, β} (E) = \int_{E} w_{α, β} d λ$ . Then $μ_{α, β}$ is a finite positive Borel measure, its relative support is all of $[- 1, 1]$ , and every moment $\int ∣ t ∣^{k} d μ_{α, β}$ is finite for $k \in N_{0}$ .

Facts & Assumptions

Given: Countable Choice and real parameters $α, β > - 1$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the Countable Choice axiom is assumed.

[F1]

A power above exponent minus one is Lebesgue integrable at zero: for every $q > - 1$ , $x \mapsto x^{q}$ on $(0, 1]$ , extended by zero at zero, is nonnegative and Lebesgue integrable with integral $1 / (q + 1)$ .

[F2]

Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume and Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable: under [A1], Borel sets in $[- 1, 1]$ lie in the Lebesgue sigma-algebra and Lebesgue measure is a measure there; an interval has measure its length (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F3]

Continuity and derivatives of positive-base real powers, 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 A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs: positive-base powers are continuous, as are the finite products and compositions used in the density.

[F4]

The measure with density $f$ relative to $μ$ : a nonnegative measurable density $w$ defines the measure $w d λ$ by $E \mapsto \int_{E} w d λ$ .

[F5]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: a continuous real function on a compact interval attains its finite minimum and maximum.

[F6]

Monotonicity and nonnegative homogeneity of the nonnegative integral: the nonnegative integral is monotone and positively homogeneous.

[F7]

A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: bounded Riemann-integrable functions on compact intervals have the same Lebesgue integral under [A1]; Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ gives the ordinary Riemann substitution on intervals away from the singular endpoints.

[F8]

Monotone convergence for the integral and Closure properties of measurable functions used by the integral: increasing nonnegative measurable cutoffs have integrals increasing to the integral of their limit.

[F9]

Riemann's criterion: a bounded $f$ on $[a, b]$ is Darboux integrable if and only if for every real $ε > 0$ there is a partition $P$ with $U (f, P) - L (f, P) < ε$ : a bounded function whose Darboux gaps can be made arbitrarily small is Riemann integrable.

Proof

1.1A1F2F3givenalgebra

The density is nonnegative and continuous on $(- 1, 1)$ by [F3]. For every real $c$ , its strict superlevel is all of $[- 1, 1]$ when $c < 0$ , and the relatively open set ${t \in (- 1, 1) : w_{α, β} (t) > c}$ when $c \geq 0$ ; thus the endpoint extension is Borel, and [F2] makes it Lebesgue measurable. Put $M_{α} = \max (1, 2^{α})$ and $M_{β} = \max (1, 2^{β})$ . On $(- 1, 0]$ the factor $(1 - t)^{α}$ is at most $M_{α}$ , and on $[0, 1)$ the factor $(1 + t)^{β}$ is at most $M_{β}$ .

2.1A1F1F6F7F8F9step 1.1constructalgebra

To justify the endpoint integral comparisons, set $w_{n} = w_{α, β} χ_{[- 1 + 1 / n, 1 - 1 / n]}$ for $n \geq 2$ . Its superlevels are Borel by step 1.1. It is bounded and continuous except possibly at the two cutoff points; isolating these in intervals of arbitrarily small total length and using uniform continuity on the remaining compact pieces proves the Darboux criterion [F9]. The same argument applies to each half-interval restriction and to the corresponding power cutoffs. Consequently [F7] and the Riemann substitutions $u = 1 + t$ and $u = 1 - t$ give $\int_{- 1}^{0} w_{n} d λ \leq M_{α} \int_{1 / n}^{1} u^{β} d λ \leq M_{α} / (β + 1)$ and $\int_{0}^{1} w_{n} d λ \leq M_{β} \int_{1 / n}^{1} u^{α} d λ \leq M_{β} / (α + 1)$ , using [F1] and [F6] for the final bounds. Since $w_{n} ↑ w_{α, β}$ pointwise, including at the endpoints, [F8] gives $\int_{[- 1, 1]} w_{α, β} d λ \leq M_{α} / (β + 1) + M_{β} / (α + 1) < \infty$ .

3.1A1F2F4F5F6step 1.1step 2.1choose

By [F4], $μ_{α, β}$ is a positive Borel measure, and step 2.1 makes its total mass finite. If $U \subseteq [- 1, 1]$ is nonempty and relatively open, choose a nondegenerate closed interval $[c, d] \subset U \cap (- 1, 1)$ . The density is continuous and strictly positive on $[c, d]$ , so [F5] gives a minimum $m > 0$ there. By [F2] and monotonicity [F6], $μ_{α, β} (U) \geq \int_{[c, d]} w_{α, β} d λ \geq m (d - c) > 0$ . Therefore every point is in the relative support.

4.1F6step 2.1algebra∎

For every $k \in N_{0}$ , $∣ t ∣^{k} \leq 1$ on $[- 1, 1]$ , so [F6] gives $\int ∣ t ∣^{k} d μ_{α, β} \leq μ_{α, β} ([- 1, 1]) < \infty$ by step 2.1. This proves all moment claims.

Depends on

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Sources

Christopher J. Bishop, The Riemann Mapping Theorem, Chapter 3 §3, Gauss-Jacobi quadrature lecture notes (standard reference, not scraped)