The shape-rate Gamma density is normalized

Given: Countable Choice and positive real parameters $a,b$.

Countable Choice is the axiom required by the bounded-Riemann-to-Lebesgue integral and Lebesgue change-of-variables suppliers. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

$\mathrm{\Gamma }(a)={\int }_0^{\mathrm{\infty }}{t}^{a-1}{e}^{-t}dt$ as a mixed improper integral. The real Gamma function by Euler's integral.

This Euler integral is finite for $a>0$ by convergence and strictly positive by the positive-integrand observation in the Gamma definition. Euler's Gamma integral converges exactly for positive real parameters The real Gamma function by Euler's integral.

Bounded Riemann-integrable functions on compact intervals have the same Lebesgue integral. A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral.

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

Monotone convergence holds for nonnegative measurable functions. Monotone convergence for the integral.

For a $C^1$ diffeomorphism between open sets, Lebesgue change of variables holds for nonnegative measurable functions under Countable Choice. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions.

Positive real powers and the real exponential are continuous on their domains. Continuity and derivatives of positive-base real powers The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$.

Under Countable Choice, one-dimensional Lebesgue measure is a complete measure on the Lebesgue sigma-algebra, which contains all Borel sets. We use its restriction to $(0,\mathrm{\infty })$. 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.