Integration of functions mapping into a vector space

[winterfors]

Given a measurable function $f$ that is not real- or complex valued, but that maps into some vector space, what are the necessary conditions for it to be integrable?

I've looked through over 20 books on integration and measure theory, but they all only deal with integration of real (or sometimes also complex) valued functions!

Can anyone point me to a reference for integration of the more general class of functions mapping onto vector spaces?

 

[micromass]

Can you tell me why you need this?? (I want some context)

Furthermore, what sigma-algebra is on the vector space??
How is the integral defined?

 

[winterfors]

The context is that I have a function
$$f:\Gamma \to {\rm K}$$
where $\Gamma$ is a set of probability measures all defined on the same sigma-algebra $\Sigma$, and ${\rm K}$ is some subset of a vector space equipped with a partial ordering.

Now, $\Gamma$ is also a probaiblity space $(\Gamma ,{\Sigma _\Gamma },{P_\Gamma })$ and I need to integrate (take the expectation of) $f$ with repsect to $P_\Gamma$

Well, all this is not really necessary to know for answering the initial question on what are the necessary conditions on $f$ in order for it to be integrable with respect to $P_\Gamma$ but there you go...

 

[Fredrik]

The book "Real and functional analysis" by Serge Lang defines integrals of Banach space-valued functions. The definition is essentially same as the one I call the "limit definition" in this post (obviously with the term "real-valued" replaced by "Banach-valued").

I wrote the definitions in the post I linked to before I understood that there's no need to say that the functions are "a.e. real-valued and measurable" right at the start. The definition is supposed to be applied to measurable functions. If a function is "integrable", it's automatically a.e. real-valued (or Banach-valued).

I have only read half a page of Lang's book to confirm that the definition does indeed apply to Banach-valued functions, but I'm going to have to read more. It looks really good.

Edit: Hm, you said "equipped with a partial ordering", but you didn't mention a norm. You will need one to use this definition.

 

[winterfors]

Tack Fredrik,

I've looked briefly at Serge Lang's book and it's along the lines of what I'm looking for. He seems to use the norm on the vector space $K$ the function injects into to define a metric to assure convergence, so maybe it would be sufficient to assume that $K$ is a subset of a metric vector space, rather than a normed one?

Hälsningar,

-Emanuel

 

[micromass]

Tack Fredrik,

I've looked briefly at Serge Lang's book and it's along the lines of what I'm looking for. He seems to use the norm on the vector space $K$ the function injects into to define a metric to assure convergence, so maybe it would be sufficient to assume that $K$ is a subset of a metric vector space, rather than a normed one?

Hälsningar,

-Emanuel

I really think you do need a norm. For example, the inequality

$$\left|\int fd\mu\right|\leq \int |f|d\mu$$

uses the norm on the Banach space. I don't think that you can say that

$$d\left(\int fd\mu,\int gd\mu\right)\leq \int d(f,g)d\mu$$

in spaces which are not normed. At least I see no way to fix the original proof (but maybe that's me).

 

[Tarantinism]

Hej!

Try to have a look at this:

http://en.wikipedia.org/wiki/Bochner_integral

Hej daº