It seems like different authors use different definitions for the space $L^\infty$. For example wikipedia starts with bounded functions, and define the seminorm $\lVert f \rVert_\infty$, and take quotients. When defining the seminorm, wikipedia takes the infimum over all $M \geq 0$ such that the set $\left\{x\in X\ |\ f(x)>M\right\}$ is null.
The books by Folland (Real Analysis) or Jones (Lebesgue Integration on Euclidean Space), on the other hand, starts with essentially bounded functions; $f$ is essentially bounded iff there is some $M\geq 0$ that makes $\left\{x\in X\ |\ f(x)>M\right\}$ a null set. Then they define the seminorm, take quotients, just like wikipedia.
The book by Cohn (Measure Theory) starts with bounded functions, but the seminorm differs! Here, the seminorm is given by the infimum over all $M\geq 0$ such that the set $\left\{x\in X\ |\ f(x)>M\right\}$ is locally null. When the given measure is $\sigma$-finite, the concept of locally null and null coincide, so this definition agrees with wikipedia's.
So the definition differs from books to books. There are at least 2*2=4 possibilities I guess;
Which seminorm do you use? ($\left\{x\in X\ |\ f(x)>M\right\}$ is null? or locally null?...)
What functions do you start with? (bounded functions? or those with finite seminorms?...)
Do the results concerning the properties of $L^p$ differs significantly when choosing a different definition for $L^\infty$ ? Do these differences affect the theory in a serious manner? Or, is it the case that, at least for familiar spaces such as $\mathbb{R}^n$ they all agree?
Thank you in advance.