Show that $f(x)=1/\sqrt x$ is measurable
Use the definition of inverse mapping: $$f^{-1}((-\infty,r])=\{x\in \mathbb{R}:f(x)\leq r\}=\begin{cases} \emptyset &r<0\\ (-\infty,0]\cup (1,\infty)& 0\leq r <1\\ (-\infty,0]\cup [1/r^2,\infty)&r\geq 1 \end{cases}$$ all these sets belong to $\mathcal{B}(\mathbb{R})$ and $\{(-\infty,r],r \in \mathbb{R}\}$ is a generator of the Borel sets so $f$ is measurable.