There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$--algebra of subsets of $S$ and and $\mu$ is a positive measure.) The two definitions are the same same in some (most?) cases (such as if $\mu$ is $\sigma$--finite) but are not always the same same. One seems to be standard in most graduate textbooks and the other is from from Dunford and Schwartz's Linear Operators, Part Part I: General Theory (abbreviated DSI). Can Can anyone recount the history and present day usage of the two definitions? For easier reference the two definitions are recalled here.
The first definition is taken from: Michael E Taylor, Measure Theory and IntegrationMeasure Theory and Integration, Graduate Graduate Studies in Mathematics, volume 76, Amer Math Soc. 2006, page 43. It is:
Definition 1.Definition 1. $L^p(S, \Sigma,\mu)$ is the set of all (equivalence classes) of $\Sigma$--measurable functions functions $f$ so that $$ \|f\|_{p}^p:= \int_S |f|^p d \mu < \infty \mbox{ if }1\le p <\infty $$$$ \|f\|_{p}^p:= \int_S \lvert f\rvert^p d \mu < \infty \text{ if $1\le p <\infty$} $$ or the essential supremum of $|f|$$\lvert f\rvert$ is finite if $p=\infty$. (The equivalence classes are for the relation $f \equiv g$ if $f=g$ $\mu$--almost everywhere.)
Here $\Sigma$-measurable is (according to DSI page 240) defined as: A function $f:S \to \mathbb{C}$ is $\Sigma$-measurable if $f^{-1}(B) \in \Sigma$ for all Borel subsets $B$ of the the complex plane $\mathbb{C}$.
The second definition is from DSI, page 119:
Definition 2.Definition 2. It is exactly the same as Definition 1 except that " “$\Sigma$--measurable''measurable” is replaced by ``“$\mu$--measurable''measurable”.
Let us recall the definition of $\mu$--measurable from DSI. First they introduce (DSI page 101) the topology associated with convergence in $\mu$ measure. A
A function $f:S \to \mathbb{C}$ is said (page 106) to be totally $\mu$--measurable if it belongs to the closure of the set of all $\Sigma$--measurable simple (complex valued) functions. A
A function $f:S \to \mathbb{C}$ is said to be $\mu$--measurable if $\chi_E f$ is totally $\mu$--measurable whenever $E \in \Sigma$ has finite $\mu$ measure. (Here $\chi_E$ denotes the characteristic function of $E$.)
In general $\Sigma$--measurable implies $\mu$--measurable but not conversely. Hence Hence Definition 1 defines a smaller set $L^p$ than Definition 2. Is it strictly smaller smaller? Not if $\mu$ is $\sigma$--finite. When then is there a difference?
Following DSI, page 296 we define $\Sigma_1$ as follows. We suppose (without loss of generality) that $(S,\Sigma,\mu)$ is complete (so that $\Sigma$ contains all subsets of sets in $\Sigma$ of measure zero). Define $\Sigma_1$ to be the family of all subsets $E$ of $S$ so that $A \cap E \in \Sigma$ whenever $A \in \Sigma$ has finite measure. Certainly $\Sigma_1 \supseteq \Sigma$ and if the containment is proper and if $E \in \Sigma_1\setminus \Sigma$ then then $\chi_E$ belongs to $L^\infty(S,\Sigma, \mu)$ as defined in DSI (second definition) but, but is not in $L^\infty(S,\Sigma, \mu)$ as defined in the first definition. Constructing Constructing an example where $\Sigma_1 \neq \Sigma$ seems to be straightforward.
Surely this is known and noted somewhere in the literature. Can anyone direct me?
