Defining similar space to $L^p$ for function of random variables

Question

We know that $L^p = \{X \mid \mathbb{E}[|X|^p] < \infty\}$ spaces are vector spaces. Let $g:\mathbb{R} \to \mathbb{R}$. I would like to define a similar space with respect to function $g$. In particular, I want something like $L_g = \{X \mid \mathbb{E}[g(X)] < \infty \}$ that makes sure $g(X)$ is integrable. However, in this definition, $L_g$ is not a vector space. Any ideas how it is possible to define such a space to assure it is a vector space?