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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$, with the identification of $\mathbb{R}$ and $\mathbb{R}^{1}$ and the notation of claims 1 and 2 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative in force; in particular each test function $\psi\in C_{c}^{\infty}(\mathbb{R})$ has a derivative $\psi'=\nabla\psi$, itself continuous and bounded. Let $\lambda_{1}$ be the Lebesgue measure on $\mathcal{B}(\mathbb{R})$.

1. (Test functions are Lebesgue integrable)¶ Every $\psi\in C_{c}^{\infty}(\mathbb{R})$ is integrable with respect to $\lambda_{1}$, and so is $\psi'$.

2. (A nonnegative test function of unit mass)¶ There is $\theta\in C_{c}^{\infty}(\mathbb{R})$ with $0\le\theta(x)$ for every $x\in\mathbb{R}$ and $\int_{\mathbb{R}}\theta\,d\lambda_{1}=1$.

3. (Test functions of large mass bounded by one)¶ For every positive real number $C$ there is $\chi\in C_{c}^{\infty}(\mathbb{R})$ with $0\le\chi(x)\le1$ for every $x\in\mathbb{R}$ and $C\le\int_{\mathbb{R}}\chi\,d\lambda_{1}$.

4. (Mean-zero correction)¶ Let $\theta$ be as in claim 2 and let $\psi\in C_{c}^{\infty}(\mathbb{R})$. Then the function $\varphi=\psi-\bigl(\int_{\mathbb{R}}\psi\,d\lambda_{1}\bigr)\theta$ belongs to $C_{c}^{\infty}(\mathbb{R})$ and satisfies $\int_{\mathbb{R}}\varphi\,d\lambda_{1}=0$.

5. (Primitive of a mean-zero test function)¶ Let $\varphi\in C_{c}^{\infty}(\mathbb{R})$ satisfy $\int_{\mathbb{R}}\varphi\,d\lambda_{1}=0$. Then there is $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\psi'(x)=\varphi(x)$ for every $x\in\mathbb{R}$.