Proof of The Value of the Gaussian Integral

Routine facts about real numbers are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, Toolkit for CA1c, The Structure of ℝ as a Field and an Ordered Ring (ix)), and so is the agreement of the earlier and current versions of the Borel $\sigma$-algebra and of the integral of nonnegative functions (The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (v)). Write $g_{0}(x)=\exp(-x^{2}/2)$, so that $c_{\mathrm{G}}=\int_{\mathbb{R}}g_{0}\,d\lambda$. The proof squares the integral, substitutes $y=xs$ in the inner integral of the resulting double integral, exchanges the order of integration, and evaluates the two one-dimensional integrals that remain.

$c_{\mathrm{G}}^{2}$ as an iterated integral. Fix $x\in\mathbb{R}$. Pulling out the constant $g_{0}(x)$ (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)) gives

$$\int_{\mathbb{R}}g_{0}(x)\,g_{0}(y)\,d\lambda(y)=g_{0}(x)\int_{\mathbb{R}}g_{0}\,d\lambda=c_{\mathrm{G}}\,g_{0}(x).$$

Now $c_{\mathrm{G}}$ is a finite nonnegative real number (The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (i)), so integrating in $x$ and pulling out the constant $c_{\mathrm{G}}$ in the same way yields $c_{\mathrm{G}}\cdot c_{\mathrm{G}}$, which is $c_{\mathrm{G}}^{2}$ (a product in $[0,\infty]$ of real numbers is their product in $\mathbb{R}$, Addition and Multiplication on [0,∞] (ii)); that is,

$$c_{\mathrm{G}}^{2}=\int_{\mathbb{R}}\Bigl(\int_{\mathbb{R}}g_{0}(x)\,g_{0}(y)\,d\lambda(y)\Bigr)d\lambda(x).$$

The substitution $y=xs$. Let $H:\mathbb{R}\times\mathbb{R}\to[0,\infty)$ be the function $H(x,s)=|x|\,g_{0}(x)\,g_{0}(xs)$. Since $\exp(u+v)=\exp u\exp v$ (Basic Properties of the Exponential Function (ii)), $g_{0}(x)\,g_{0}(xs)=\exp(-(1+s^{2})x^{2}/2)$, that is,

$$H(x,s)=|x|\exp\bigl(-(1+s^{2})x^{2}/2\bigr)\qquad(x,s\in\mathbb{R}).$$

With the product $\sigma$-algebra $\otimes$ and the product measure as in Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, the nonnegative function $H$ lies in $\mathcal{M}^{+}(\mathbb{R}\times\mathbb{R},\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}))$, being built from the coordinate projections by products, the absolute value and composition with $g_{0}$, which is measurable as $g_{0}\in\mathcal{L}^{1}(\lambda)$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (vi), Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii), The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (i), Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)). Since $\lambda$ is a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (v); these facts, like several results cited below, rest on dependent choice), Tonelli's theorem (Tonelli's and Fubini's Theorems (i)) shows that the map $x\mapsto\int_{\mathbb{R}}H(x,s)\,d\lambda(s)$ lies in $\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

Now fix $x\neq0$. The function $y\mapsto g_{0}(x)\,g_{0}(y)$ lies in $\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$ by homogeneity (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)). The affine change of variables (Affine Change of Variables for Lebesgue Measure on the Real Line (ii), with $a=x$ and $b=0$), which replaces $y$ by $xs$ at the cost of the factor $|x|$, followed by pulling the constant $|x|$ inside the integral (homogeneity again), gives

$$\int_{\mathbb{R}}g_{0}(x)\,g_{0}(y)\,d\lambda(y)=|x|\int_{\mathbb{R}}g_{0}(x)\,g_{0}(xs)\,d\lambda(s)=\int_{\mathbb{R}}H(x,s)\,d\lambda(s).$$

So the two inner integrals, as functions of $x$ in $\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (by $c_{\mathrm{G}}^{2}$ as an iterated integral and the previous paragraph), can differ only at $x=0$. Since $\lambda(\{0\})=0$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)), they have the same integral (The Lebesgue Integral and Null Sets (ii)), and therefore

$$c_{\mathrm{G}}^{2}=\int_{\mathbb{R}}\Bigl(\int_{\mathbb{R}}H(x,s)\,d\lambda(s)\Bigr)d\lambda(x).$$

Exchanging the order. By Tonelli's theorem (Tonelli's and Fubini's Theorems (i)), both iterated integrals of the nonnegative measurable function $H$ equal its integral $\int H\,d(\lambda\otimes\lambda)$ with respect to the product measure, so

$$c_{\mathrm{G}}^{2}=\int_{\mathbb{R}}\Bigl(\int_{\mathbb{R}}H(x,s)\,d\lambda(x)\Bigr)d\lambda(s).$$

The inner integral. Fix $s\in\mathbb{R}$ and put $a=1+s^{2}$, a positive real number. By the form of $H$ displayed in The substitution $y=xs$, $H(x,s)=|x|\exp(-ax^{2}/2)$ for every $x$, so The Integral of |x| exp(−ax²/2) over ℝ (i) gives

$$\int_{\mathbb{R}}H(x,s)\,d\lambda(x)=\frac{2}{1+s^{2}}\qquad(s\in\mathbb{R}).$$

The outer integral. Substituting The inner integral into Exchanging the order, pulling out the constant $2$ (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)) and using The Integral of 1/(1+s²) over ℝ (i), we obtain

$$c_{\mathrm{G}}^{2}=\int_{\mathbb{R}}\frac{2}{1+s^{2}}\,d\lambda(s)=2\int_{\mathbb{R}}\frac{1}{1+s^{2}}\,d\lambda(s)=2\pi .$$

Conclusion. By The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (i), $c_{\mathrm{G}}$ is a positive real number, and by The outer integral its square is $2\pi$. Hence $c_{\mathrm{G}}$ is the nonnegative square root of $2\pi$ (The k-th Root and the Square Root of a Nonnegative Real Number (ii), Properties of k-th Roots: Inverse to Powers, Absolute Values, Products, Monotonicity, Subadditivity, Differences and Limits (i)), that is, $c_{\mathrm{G}}=\sqrt{2\pi}$, which proves (i).