5
$\begingroup$

In measure theory one usually starts with a $\sigma$-algebra $A$ of sets and considers a measure $\mu:A\to [0,\infty]$. I'm interested in abstracting this definition to allow more general domains and codomains for the measure function. I'm aware of measures allowed to take values in $\mathbb {R}$ as well as in $\mathbb {C}$ and I'm also aware of vector valued measures. I'm not really familiar with measures having a domain that is anything but a $\sigma$-algebra of sets (except of course for finitely additive measures).

So, any information or reference to work done along these lines would be greatly appreciated. Especially, for the most general case where the domain of $\mu$ is allowed to be any $\sigma$-complete lattice and the codomain is the most general kind of lattice (probably complete with some binary operation $+$) that will support a good theory. But also non so far-reaching generalizations would be great. Thanks.

  • 0
    my point I guess is that *general measure theory* is not that general if it still requires the values of the measure to be real numbers (or closely related structures such as the complex numbers of vector spaces over C). The distinction between an abstract measure space and a concrete one is that in the former the domain of the measure is a sigma algebra of sets. I'm asking for a further generalization by allowing the codomain of the measure to be more general than R.2012-11-05

1 Answers 1

3

Heinz König did a lot of work on measures defined on lattices and he has written a very demanding book on these issues, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Ultimately, one can extend these measures to $\sigma$-algebras though, König is mainly concerned with extension procedures and regularity properties as far as I can tell.

  • 0
    Thank you Michael, great reference exactly along the lines I was looking for.2012-11-05