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Assume I have a measurable space $(\Omega, \mathcal{F}, \mu)$, $\mu$ being a finite measure ($\mu(\Omega) < \infty$, if needed). Let $L = \{F:L^1(\Omega, \mu) \mapsto \mathbb{R}\}$ ($L$ is the set of not necessarily linear functionals on my measurable functions).

My questions are:

  1. Can we define a $\sigma$-algebra on $L$? I guess at least the power set should do
  2. Can we define a measure $\nu$ on $L$? I've started thinking of $\forall f \in L^1(\Omega, \mu),~\nu(\{f\}=\|f\|_{L^1(\Omega,\mu)})$. But then I am pretty sure we lose the finiteness of the measure
  3. Do we have any results such as Hardy and littlewood inequalities?

They might be other ways of defining all this lot. And actually, I might forget an important part of the whole thing, please do tell me if there is anything completely stupid in there. And, of course, thanks for any help you can give!

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You can always put the trivial $\sigma$-algebra $\{L,\emptyset\}$ and the measure $\nu(L)=1$, $\nu(\emptyset)=0$ on it. I guess you want something more useful. – Michael Greinecker Mar 8 '12 at 12:45
The measure you define is by extension defined on the $\sigma$-algebra $\{A \subseteq L \mid A \text{ or } L\setminus A \text{ is countable}\}$ and it's finite exactly on finite sets. – martini Mar 8 '12 at 14:04
Thanks guys, at least I have something to start with. Do you know any other of defining usefull (indeed Michael, I'not going to go far with yours! :) ) measures on such spaces? I'll eventually let you know how it goes, IF it goes any further! – Jean-Luc Bouchot Mar 8 '12 at 22:07

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