 Solution

Lebesgue measurability of $g$ means that $g^{-1}(U)$ is a Lebesgue measurable subset of $\mathbb{R}$ for every open subset $U\subseteq \mathbb{R}$. Equivalently one can check all sets $\{x:g(x)>t\}$ with real $t$.

There are two conventions for a step function on the whole real line. Under the bounded-support integration convention it is a finite linear combination of indicators of bounded intervals; under the unrestricted piecewise-constant convention nonzero constant tails are allowed. The conclusions below state both conventions explicitly, since only the first assertion depends on this choice.