Proof of The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments

Throughout, the axiom of dependent choice is assumed, as the results on $\lambda$ cited below require; $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv)); continuous maps $\mathbb{R}\to\mathbb{R}$ are measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)); $n$ ranges over $\mathbb{N}$, so $n>0$ in $\mathbb{R}$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi)) and $-n<n$; and for $a<b$ and $g:\mathbb{R}\to\mathbb{R}$ with $g|_{[a,b]}$ continuous we write $\int_{a}^{b}g$ for the Riemann integral of $g|_{[a,b]}$, and use that $g\mathbf{1}_{[a,b]}\in\mathcal{L}^{1}(\lambda)$ with $\int_{\mathbb{R}}g\mathbf{1}_{[a,b]}\,d\lambda=\int_{a}^{b}g$ (The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (i)). For a continuous $h$ on $\mathbb{R}$ we write $I_{n}(h)=\int_{-n}^{n}h$; by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (i) and linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)), $I_{n}$ is linear in $h$.

Step 1 (derivatives and bounds). The map $q(x)=-x^{2}/2$ is differentiable on the interval $\mathbb{R}$ with $q'(x)=-x$ (Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v)), so $g_{0}=\exp\circ q$ is differentiable with $g_{0}'(x)=-x\exp(-x^{2}/2)$ (Basic Properties of the Exponential Function (vii), Chain Rule for Real Functions on Intervals (i)). Hence, by Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), $G=-g_{0}$ satisfies $G'=g_{1}$, and $g_{1}(x)=x\,g_{0}(x)$ and $g_{2}(x)=x\,g_{1}(x)$ are differentiable on $\mathbb{R}$, with, by the product rule and the sum rule (Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv)),

$$G'(x)=g_{1}(x),\qquad g_{1}'(x)=g_{0}(x)+x\,g_{0}'(x)=g_{0}(x)-g_{2}(x)\qquad(x\in\mathbb{R}).$$

So $g_{0},g_{1},g_{2}$ and $G$ are continuous on $\mathbb{R}$ (The Increment Form of the Derivative, and Differentiability Implies Continuity (ii)), and for $a<b$ the restriction of a map differentiable on $\mathbb{R}$ to $[a,b]$ is differentiable there with the restricted derivative, and the restriction of a continuous map is continuous (Restriction Stability of Continuity and of the Derivative (ii), Restriction Stability of Continuity and of the Derivative (i)). Since $\exp$ is positive with $\exp(0)=1$ (Basic Properties of the Exponential Function (iii), Basic Properties of the Exponential Function (i)), we have $g_{0}>0$, $g_{2}\ge0$ and $g_{0}(0)=1$. Finally $g_{0}(x)=1/\exp(x^{2}/2)$ and $\exp(x^{2}/2)\ge1+x^{2}/2$ (Basic Properties of the Exponential Function (iv), Basic Properties of the Exponential Function (v)), so

$$g_{0}(x)\bigl(1+x^{2}/2\bigr)\le1,\qquad\text{hence}\qquad g_{0}(x)\le1\quad\text{and}\quad x^{2}g_{0}(x)\le2\qquad(x\in\mathbb{R}).$$

Step 2 (integrals over compact intervals). Let $F:\mathbb{R}\to\mathbb{R}$ be differentiable with continuous derivative $f$, and let $a<b$. Then $f|_{[a,b]}$ is bounded and integrable (A Continuous Function on a Closed Interval is Riemann Integrable (i)), $F|_{[a,b]}$ is continuous with derivative $f$ on $(a,b)$ by Step 1, and the fundamental theorem of calculus (Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative (i)) gives $\int_{a}^{b}f=F(b)-F(a)$. Applied to $F=G$, to $F=g_{0}$ (with $f=-g_{1}$) and to $F(t)=t$ (with $f$ the constant $1$, continuous as a differentiable map, Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i)), this gives

$$\int_{0}^{n}g_{1}=1-\exp(-n^{2}/2)=\int_{-n}^{0}(-g_{1}),\qquad\int_{-n}^{n}g_{1}=G(n)-G(-n)=0,\qquad\int_{-1}^{1}1=2 .$$

Step 3 ((i)). For every $n$ we have, pointwise on $\mathbb{R}$,

$$g_{0}\mathbf{1}_{[-n,n]}\le\mathbf{1}_{[-1,1]}+g_{1}\mathbf{1}_{[0,n]}+(-g_{1})\mathbf{1}_{[-n,0]} .$$

Indeed all three terms on the right are nonnegative; on $[-1,1]$ the left side is at most $1$ by Step 1; for $1\le x\le n$ it is $g_{0}(x)\le x\,g_{0}(x)=g_{1}(x)$; for $-n\le x\le-1$ it is $g_{0}(x)\le-x\,g_{0}(x)=-g_{1}(x)$; and for $|x|>n$ it is $0$. All four functions lie in $\mathcal{L}^{1}(\lambda)$ (The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (i), for $\mathbf{1}_{[-1,1]}$ with $g\equiv1$), so linearity and monotonicity of the integral (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ix)) and Step 2 give

$$I_{n}(g_{0})\le2+2\bigl(1-\exp(-n^{2}/2)\bigr)\le4 .$$

Hence $g_{0}\in\mathcal{L}^{1}(\lambda)$ by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (iii), and $c_{\mathrm{G}}=\sup_{n}I_{n}(g_{0})\le4$ by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (ii), the supremum being taken in $[0,\infty]$ (Order and Suprema in the Extended Real Line (iii)). So $c_{\mathrm{G}}$ is a real number, equal to the integral of $g_{0}$ as an element of $\mathcal{L}^{1}(\lambda)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)), and $I_{n}(g_{0})\le c_{\mathrm{G}}$ for every $n$. For positivity, $g_{1}\mathbf{1}_{[0,1]}\le g_{0}$ on $\mathbb{R}$, since $x\,g_{0}(x)\le g_{0}(x)$ for $0\le x\le1$ and the left side vanishes elsewhere; and $\exp(-1/2)=g_{0}(1)\le2/3<1$ by the bound of Step 1. So monotonicity and Step 2 give

$$c_{\mathrm{G}}\ge\int_{0}^{1}g_{1}=1-\exp(-1/2)>0 .$$

Step 4 (the integral of $g_{2}$). By Step 1, $g_{1}$ is differentiable with the continuous derivative $g_{0}-g_{2}$, so Step 2 with $F=g_{1}$ and the linearity of $I_{n}$ give

$$I_{n}(g_{0})-I_{n}(g_{2})=g_{1}(n)-g_{1}(-n)=2n\exp(-n^{2}/2).$$

Thus $I_{n}(g_{2})\le I_{n}(g_{0})\le c_{\mathrm{G}}$, so $g_{2}\in\mathcal{L}^{1}(\lambda)$ by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (iii), and by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (iv)

$$I_{n}(g_{2})\to\int_{\mathbb{R}}g_{2}\,d\lambda,\qquad I_{n}(g_{0})\to c_{\mathrm{G}} .$$

Since $I_{n}(g_{2})\le c_{\mathrm{G}}$ for every $n$, Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x) gives $\int_{\mathbb{R}}g_{2}\,d\lambda\le c_{\mathrm{G}}$. Conversely, the bound $x^{2}g_{0}(x)\le2$ of Step 1 gives

$$0\le2n\exp(-n^{2}/2)=\frac{2}{n}\,n^{2}g_{0}(n)\le\frac{4}{n}.$$

Let $\varepsilon>0$. Choose $N_{1}\in\mathbb{N}$ with $I_{n}(g_{0})>c_{\mathrm{G}}-\varepsilon/2$ for every $n\ge N_{1}$, and, by The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (vii) applied to $8/\varepsilon$, an $N_{2}\in\mathbb{N}$ with $n>8/\varepsilon$, that is $4/n<\varepsilon/2$, for every $n\ge N_{2}$. For $n\ge\max(N_{1},N_{2})$ then $I_{n}(g_{2})\ge I_{n}(g_{0})-4/n>c_{\mathrm{G}}-\varepsilon$, so $\int_{\mathbb{R}}g_{2}\,d\lambda\ge c_{\mathrm{G}}-\varepsilon$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x). As $\varepsilon$ was arbitrary, $\int_{\mathbb{R}}g_{2}\,d\lambda=c_{\mathrm{G}}$.

Step 5 (the integral of $g_{1}$). Since $|x|\le1+x^{2}$, we have $|g_{1}|\le g_{0}+g_{2}$, and $g_{0}+g_{2}\in\mathcal{L}^{1}(\lambda)$ by Steps 3 and 4 (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)); as $g_{1}$ is measurable, $g_{1}\in\mathcal{L}^{1}(\lambda)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v)). Hence $h=g_{0}+g_{1}+g_{2}$ lies in $\mathcal{L}^{1}(\lambda)$, with

$$\int_{\mathbb{R}}h\,d\lambda=2c_{\mathrm{G}}+\int_{\mathbb{R}}g_{1}\,d\lambda$$

by linearity and Step 4. Moreover $h(x)=(1+x+x^{2})\,g_{0}(x)\ge0$, as $1+x+x^{2}=(x+1/2)^{2}+3/4>0$, and $h$ is continuous by Step 1; so $I_{n}(h)\to\int_{\mathbb{R}}h\,d\lambda$ by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (iv). On the other hand $I_{n}(g_{1})=0$ by Step 2, so the linearity of $I_{n}$ and Step 4 give

$$I_{n}(h)=I_{n}(g_{0})+I_{n}(g_{2})\to2c_{\mathrm{G}}$$

(Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi)). By uniqueness of limits (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (ii)), $2c_{\mathrm{G}}+\int_{\mathbb{R}}g_{1}\,d\lambda=2c_{\mathrm{G}}$, so $\int_{\mathbb{R}}g_{1}\,d\lambda=0$. Steps 4 and 5 prove (ii).