I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also using (2) the positive definiteness of the kernel matrix $K$ constructed from $\phi$, respectively in Theorem 2.5.2 and 2.5.3. The theorems go as follows:

**Theorem 2.5.2: (Hausdorff-Bernstein-Widder theorem: Laplace transform characterization of completely monotone functions)**

A function $\phi: [0,\infty) \to \mathbb{R}$ is completely monotone if and only it is the Laplace transform of a finite non-negative Borel measure $\mu$ on $[0,\infty)$, i.e. $\phi$ is of the form:

$$\phi(r)= \mathcal{L}\mu(r)=\int_{0}^{\infty} e^{-rt}d\mu(t)$$

**Theorem 2.5.3:** **(Theorem relating completely monotone functions and positivity of kernel matrix with radially symmetric kernels)**

A function $\phi$ is completely monotone on $[0,\infty)$ if and only if $\Phi(x):=\phi(||x||^2)$ is positive definite and radial on $\mathbb{R}^d$, $ \forall d \in \mathbb{N}$.

**My question is: are the following true regarding strict positivity of kernel matrices and the completely monotone function being non-constant?**

*Guess 1:*

A **non-constant** function $\phi: [0,\infty) \to \mathbb{R}$ completely monotone if and only it is the Laplace transform of a finite non-negative Borel measure $\mu$ on $[0,\infty)$ **not of the form** $c\delta_0, c>0$, i.e. $\phi$ is of the form:

$$\phi(r)= \mathcal{L}\mu(r)=\int_{0}^{\infty} e^{-rt}d\mu(t)$$ where $\mu$ is **not of the form** $c\delta_0, c>0$

*Guess 2:*

A **non-constant** function $\phi$ is completely monotone on $[0,\infty)$ if and only if $\Phi(x):=\phi(||x||^2)$ is **strictly** positive definite and radial on $\mathbb{R}^d \forall d \in \mathbb{N}$?

N.B. It's worthwhile mentioning that in this book and in the relevant literature, positive definite normally means positive semidefinite elsewhere, and strictly positive definite normally means positive definite elsewhere.