When I first learned measure theory I was told that $\mathcal L^\infty(X,\mathcal A,\mu,\mathbb K)$ is the set of all measurable $f:X\to\mathbb K$ (where $\mathbb K$ is either the real or the complex numbers) that are essentially bounded, i.e. such that $|f|\leq C$ $\mu$-a.e. for some $C\geq0$. One then defines the seminorm $\Vert f\Vert_\infty$ as the infimum of such $C$ (the essential supremum) and $L^\infty$ as the quotient of $\mathcal L^\infty$ wrt $\{\Vert f\Vert_\infty=0\}$; this is a Banach space and if $\mu$ is $\sigma$-finite, then $\varphi\mapsto(f\mapsto\int_Xf\varphi\, d\mu)$ is an isometric isomorphism $L^\infty\to(L^1)'$.
Recently I've encountered a modified definition of $L^\infty$ that allows for this duality isomorphism to hold in more general cases. E. Behrends in his text "Maß- und Integrationstheorie" uses the following definitions:
- Let $\mathcal F\subset\mathcal A$ be the collection of measurable subsets of finite measure. A subset $N\subset X$ (not assumed to be $\mathcal A$-measurable) is called locally null if $N\cap A\in\mathcal A$ and $\mu(N\cap A)=0$ for all $A\in\mathcal F$; a property is said to hold locally-a.e. if it holds on the complement of a locally null set.
- A function $f:X\to[-\infty,+\infty]$ is called locally measurable if $f\chi_A:A\to[-\infty,+\infty]$ is measurable (wrt the trace algebra on $A$) for all $A\in\mathcal F$. The function is called locally essentially bounded if $|f|\leq C$ locally-a.e. for some $C\geq0$.
- $\mathcal L^\infty_0(X,\mathcal A,\mu,\mathbb K)$ is declared as the space of all locally measurable locally essentially bounded functions $X\to\mathbb K$. The locally essential supremum (the infimum of the above $C\geq0$) is then a seminorm on $\mathcal L^\infty_0$ and the quotient space $L^\infty_0$ is a Banach space.
If $\mu$ is $\sigma$-finite, then $L^\infty_0=L^\infty$ and nothing new is introduced. With these definitions the "weighted integration" map $L^\infty_0\to(L^1)'$ is once again an injective isometry and Behrends claims it is surjective if and only if the measure space $(X,\mathcal A,\mu)$ satisfies the following criterion called "localizability":
- For any collection of measurable functions $f^A:A\to\Bbb R$ indexed by $\mathcal F$ such that $$A,A'\in\mathcal F\implies f^A_{A\cap A'}=f^{A'}_{A\cap A'}\text{ $\mu$-almost everywhere}$$ (presumably on $A\cap A'$, the text isn't clear on this), there exists a locally measurable $f:X\to\Bbb R$ such that $f|_A=f^A$.
My questions are the following:
- How canonical are these definitions/results? Cohn and Folland in their texts define $\mathcal L^\infty_0$ similarly, but with certain deviations, and localizability isn't even mentioned. No other text I'm familiar with addresses these concepts.
- In what ways -- besides the duality statement -- does $L^\infty_0$ differ from "the usual" $L^\infty$? Do either of the spaces have any desirable properties that the other doesn't?
- Can the Radon-Nikodym theorem be generalized to localizable spaces (if so, how) or do we need the stronger notion of decomposability?
Edit: I have seen the linked post and neither of the answers there actually address my questions, so I ask that my post be unlocked.
Edit2: I don't understand what the moderators' issue is with this post. There are two answers in the linked post, neither is accepted and there are no elucidating comments that might be of use. The first answer explicitly says that they do not know the answer to the question and only briefly mentions locally null sets. The second answer only repeats the definition of locally null sets and how for $\sigma$-finite spaces this reduces to measurable null sets. Neither answer in the linked post mentions local measurability or localizability or the above definition of $L^\infty_0$ in any capacity and neither answer addresses any of my 3 questions. I once again ask that this post be unlocked.
