Suppose $x_0 =0$. A particle moves as follows:
$$x_t = \int_0^t a(s) ds$$
where $a: \mathbb{R}_+ \to \{-1,0,1\}$ is a measurable function.
Suppose, I have that $a_s = 1$ if $x_s = 0$. I want to write the following, seemingly intuitive, claim.
Claim: $x_s \ge 0$ for all $s \ge 0$.
The logic is obvious. The moment the particle hits $0$, it's pushed to the right. But I just don't have a way to argue this formally. How can I proceed?
Thanks.