The Beta density has finite positive Lebesgue integral

Given: Countable Choice and positive real numbers $\alpha ,\beta$.

Countable Choice is the axiom assumed by the bounded-Riemann-to-Lebesgue integral theorem. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

The real Beta function is the improper integral of $p^{\alpha -1}(1-p{)}^{\beta -1}$ on $(0,1)$. Euler's real Beta integral.

For positive $\alpha ,\beta$, $B(\alpha ,\beta )=\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )\mathrm{/}\mathrm{\Gamma }(\alpha +\beta )$. The real Beta--Gamma identity.

The Euler Gamma integral converges to a finite real value for every positive argument. Euler's Gamma integral converges exactly for positive real parameters.

A bounded Riemann-integrable function on a closed bounded interval is Lebesgue measurable and has the same Lebesgue and Riemann integrals under Countable Choice. A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral.

Continuous functions on a closed bounded interval are bounded and Riemann integrable. A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion.

Increasing limits of nonnegative measurable functions have integrals equal to the increasing limit of their integrals. Monotone convergence for the integral.

The improper Beta integral converges for positive parameters. Euler's Beta integral converges exactly for two positive parameters.

Positive real powers are continuous on $(0,\mathrm{\infty })$. Continuity and derivatives of positive-base real powers.

Under Countable Choice, the stated Lebesgue set function is a complete measure on the Lebesgue sigma-algebra, which contains every Borel set. We use its restriction to $(0,1)$. Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$ Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable.

For $s>0$, $\mathrm{\Gamma }(s)$ is the improper integral ${\int }_0^{\mathrm{\infty }}{t}^{s-1}{e}^{-t}dt$, with separate improper ends at zero and infinity. The real Gamma function by Euler's integral.

For $t>0$, $t^r=\exp (r\log t)$. Real powers for positive bases, with the zero-base positive-exponent convention.

The logarithm is strictly increasing on $(0,\mathrm{\infty })$ and $\log 1=0$. Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm.

The exponential is continuous and strictly increasing on $\mathbb{R}$, and $\exp (x)>0$ for every real $x$. The exponential function is strictly increasing, The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$.

Proper Riemann integration preserves nonnegativity and pointwise order; if an integrable function is bounded below by $c$ on $[a,b]$, its integral is at least $c(b-a)$. If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$.

Splitting a convergent improper integral at a nonsingular finite point changes its value by the corresponding proper integral and preserves convergence of the remaining tail. Improper convergence is independent of finite truncations and split points.