Consider the definition of the Lebesgue integral for a positive function $X\rightarrow [0,+\infty]$:
$$ \int f(x) d\mu=\sup_{g\in S, \forall x : g(x)\leq f(x)} \left(\int g(x) d\mu \right)$$
where $X$ is a general measure space and $S$ the set of all simple functions $X\rightarrow\mathbb{R}$.
This definition, I think, is valid for all function $f$. Why do then most of the textbooks consider only measurable $f$?
It is exactly as you say, one can define the integral for all functions. However, it won't have some nice property, such as linearity.