In Theorem XI.3 of Scattering Theory by Reed and Simon, the authors define two sets, $N_{+}$ and $N_{-}$, as follows: \begin{align*} \ddot{r}(t) &= F(r(t)) \\ N_{+} &= \{\langle q, v \rangle | \overline{\lim_{t \to +\infty}} |r_{q,v}(t)| < \infty\} \\ N_{-} &= \{\langle q, v \rangle | \overline{\lim_{t \to -\infty}} |r_{q,v}(t)| < \infty\} \end{align*} Here $F = -\nabla V$ and $V \to 0$ as $r \to 0$, furthermore F obeys some constraint regarding boundedness and decaying conditions namely: \begin{align} |F(r)| &\leq C|r|^{-a} \tag{3a} \\ |F(r) - F(r')| &\leq D_R |r-r'|, \quad |r-r'|<1, \quad |r|<R \tag{3b} \\ |F(r)| &\leq Cr^{-a}, \quad \alpha > 2 \tag{4a} \\ |F(x) - F(y)| &\leq Dr^{-\beta} |x-y|, \quad \text{for all } |x|, |y| > r \text{ and } \beta >2 \tag{4b} \end{align}
The theorem deals with asymptotic completeness in two-body classical particle scattering. The sets N+ and N- represent the collection of initial conditions (position and velocity) for which the solutions of the classical equations of motion remain bounded as t approaches positive or negative infinity, respectively. I'm not very keen on measurability of sets and I just know the theoretical basis but I'd really appreciate if someone could provide a proof that $N_{+}$ and $N_{-}$ are measurable sets since I wouldn't even know where to start from. Thank you in avance.