In these measure theory notes, $L^p$ spaces are discussed. Right after the title of the section "7.1 $L^p$ spaces", it is stated that, "For definiteness, we consider real-valued functions. Analogous results apply to complex-valued functions."
I am a little confused by this statement. What does "definiteness" refer to with regards to real-valued functions? How does this property of real-valued functions versus complex-valued functions simplify things?
I suspect that what this is about is that the author wishes not to confuse the discussion of the concepts of $L^p$ spaces by having to pay careful attention to the notational differences between complex valued functions and real valued functions, and some of the implications of those notational issues which get in the way of the central concepts of $L^p$ spaces. The author believes it is possible to explain the concepts of $L^p$ spaces in a simpler and more straightforward manner by avoiding those issues.