This semester I am taking a course on Measure and Integration Theory. Our professor provided us with a script, written by him that covers the material of the course and contains the proofs. Nontheless there is another professor at my university that is a lot more rigurious and precise (which fits my needs more) and now I am parallel reading his book on Measure and Integration Theory as well. Up until the chapter about $L^p$ spaces everything was very similar.
But then my professor defined $L^p$ spaces using functions (equivalence classes of functions) that map from a set $\Omega$ to the extended real line $\overline{\mathbb{R}}$, where as the book I am also working with defined them as functions (again taking the equivalence class) that map from a set $\Omega$ to the real line $\mathbb{R}$. Now my question is, is there an argument to define them in one of the above ways? Are the definitions related in some sense? And will the theory that comes out of the definitions be slightly different?
The book I mentioned above having the definitions with $\mathbb{R}$ can be found here since the professor made the pdf publicly available on his homepage. And the definitions can be found on page 115.
This question was bugging me for so long. Thanks a lot in advance.