Suppose $f \geq 0$ on $\mathbb{R}$ be monotonically non-decreasing. Let $T_f$ be the distribution given by $f$. Then, $T_f \geq 0$ for all $\psi \in C^{\infty}_c(\mathbb{R})$ with $\psi(x) \geq 0$ on its support.
Is it then true that $T'_f$ is also a positive distribution?
As a side question, how does one prove that a function has distributional derivative given by a measure if $T_f$ is not positive? Is it possible for $T'_f$ to be given by a measure if $T'_f$ is not positive?
The question can actually be answered in the context of Riemann-Stieltjes integrals: Note that we have the following integration by part formula, when $f$ is nondecreasing and $\phi$ is of bounded variations:
$$\int_a^b f d\phi = f(b)\phi(b) - f(a)\phi(a) - \int_a^b \phi df.$$
we choose $a, b$ large enough so that $\phi(a) = \phi(b) = 0$. Thus
$$T_f'(\phi) = -\int f \phi'dx = -\int f d\phi = \int \phi df.$$
Since $f$ is nondecreasing, $T_f'$ is positive.