Proof of Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Let $\iota$ denote in this proof the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and never the concatenation map of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. By claim 2 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative every $\psi\in C_{c}^{\infty}(\mathbb{R})$ has a derivative $\psi'$, continuous and bounded, and $\psi$ itself is continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous.

Step 1 (Claim 1). Let $\psi\in C_{c}^{\infty}(\mathbb{R})$. It is continuous with compact support, so A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable gives that it is bounded and integrable with respect to $\lambda_{1}$.

For $\psi'$, note that $\psi$ vanishes on the complement of its support, an open set by Closed Subset of a Topological Space, the support being closed; hence $\psi'$ vanishes there too, so the set where $\psi'$ is nonzero is contained in the support of $\psi$ and the support of $\psi'$, being the closure of that set, is a closed subset of the compact support of $\psi$, hence compact by Closed Subset of a Compact Space is Compact. As $\psi'$ is continuous, A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable applies to it as well.

Step 2 (Claim 3). Let $C$ be a positive real number. By claim 1 of The Archimedean Property of the Real Numbers there is $n\in\mathbb{N}$ with $C<\kappa_{1}\iota(n)$, where $\kappa_{1}$ is the positive real number of claim 2 of The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$; here we use that $C\kappa_{1}^{-1}<\iota(n)$ for some $n$ and multiply by the positive $\kappa_{1}$, by claim 10 of Elementary Order Arithmetic in an Ordered Field.

By Existence of Smooth Bump Functions on Euclidean Space, applied in dimension $1$ with $x_{0}=0$, $r=\iota(n)$ and $s=\iota(n)+1$, there is a smooth $\chi:\mathbb{R}\to\mathbb{R}$ with $0\le\chi\le1$, with $\chi=1$ on $\bar{B}(0,\iota(n))$ and with $\chi=0$ off $\bar{B}(0,\iota(n)+1)$. Its support is a closed subset of the compact set $\bar{B}(0,\iota(n)+1)$, compact by Heine-Borel Theorem in $\mathbb{R}^n$, hence compact by Closed Subset of a Compact Space is Compact; so $\chi\in C_{c}^{\infty}(\mathbb{R})$. Since $\mathbf{1}_{\bar{B}(0,\iota(n))}\le\chi$ pointwise, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set give

the middle identity by claim 3 of The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$.

Step 3 (Claim 2). By Step 2, applied with $C=1$, there is $\chi\in C_{c}^{\infty}(\mathbb{R})$ with $0\le\chi\le1$ and $1\le c$, where $c=\int_{\mathbb{R}}\chi\,d\lambda_{1}$; in particular $c$ is positive, since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field and claim 2 of that lemma then gives $0<c$. Put $\theta=c^{-1}\chi$, a test function by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §linear, nonnegative because $c^{-1}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and with $\int_{\mathbb{R}}\theta\,d\lambda_{1}=c^{-1}c=1$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral.

Step 4 (Claim 4). Let $\theta$ be as in claim 2 and $\psi\in C_{c}^{\infty}(\mathbb{R})$, and put $a=\int_{\mathbb{R}}\psi\,d\lambda_{1}$, a real number by claim 1. Then $\varphi=\psi-a\theta$ belongs to $C_{c}^{\infty}(\mathbb{R})$ by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §linear, and by claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

Step 5 (Claim 5). Let $\varphi\in C_{c}^{\infty}(\mathbb{R})$ with $\int_{\mathbb{R}}\varphi\,d\lambda_{1}=0$. Its support is compact, hence bounded by Heine-Borel Theorem in $\mathbb{R}^n$, so there is a real number $R$ with $1<R$ such that $\varphi(t)=0$ whenever $R-1\le|t|$.

The function $\varphi$ is continuous, hence Riemann integrable on every closed interval by A Continuous Function on a Closed Interval is Riemann Integrable. Define $\psi:\mathbb{R}\to\mathbb{R}$ by

The two prescriptions agree at $x=-R$, where the integral is $0$.

The derivative. Let $x\in\mathbb{R}$. If $-R<x$, apply claim 3 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval on the interval $[-R,x+1]$, on which $\varphi$ is continuous: the function $u\mapsto\int_{-R}^{u}\varphi$ is differentiable at $x$ with derivative $\varphi(x)$, and it agrees with $\psi$ on $[-R,x+1]$. If $x\le-R$, then $x<-R+1$ and $\varphi$ vanishes on $(-\infty,-R+1)$, so $\psi$ vanishes on that open set, whence $\psi$ is differentiable at $x$ with $\psi'(x)=0=\varphi(x)$. Hence $\psi'=\varphi$ everywhere.

Compact support. For $x<-R+1$ one has $\psi(x)=0$, as just noted, and also $\psi(-R+1)=0$, since $\varphi$ vanishes on $[-R,-R+1]$ and the Riemann integral of the zero function vanishes. For $R-1\le x$, the function $\varphi$ vanishes on $[R-1,x]$, so by Additivity of the Riemann Integral on Adjacent Intervals and Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval,

the middle identity because $\varphi$ vanishes outside $[-R,R-1]$. Therefore the set where $\psi$ is nonzero is contained in the bounded set $\{x:|x|\le R\}$, and the support of $\psi$, its closure, is closed and bounded, hence compact by Heine-Borel Theorem in $\mathbb{R}^n$.

Smoothness. Let $D$ consist of $\psi$ together with $\varphi$ and all its iterated derivatives. Every member of $D$ is differentiable at every point of $\mathbb{R}$: $\psi$ by the previous paragraph, and the iterated derivatives of $\varphi$ because $\varphi$ is smooth, by Multi-Index Partial Derivatives of a $C^k$ Map and of a Smooth Map together with claim 2 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. The family $D$ is closed under taking derivatives, since $\psi'=\varphi\in D$ and the derivative of an iterated derivative of $\varphi$ is again one. Hence claim 3 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line gives that every member of $D$, in particular $\psi$, is smooth on $\mathbb{R}$.

Thus $\psi$ is smooth with compact support, so $\psi\in C_{c}^{\infty}(\mathbb{R})$, and $\psi'=\varphi$.