Premise 1: my source is the Rudin - Functional Analysis.
Premise 2: i'm not a mathmo so forgive for the mistakes
A couple of question on the subject... An example of Banach Algebra is the set of the bounded linear operators $\cal{B}(X)$ where $X$ is a banach space.
On the definition of integral (section 10.22) it says
If $A$ is a Banach algebra and $f$ is a continous $A$-valued function on some compact Hausdorf space $Q$ on which a complex Borel measure $\mu$ is defined then $\int f d\mu$ exists...
I'm just trying to find some example of such integral (i.e. i want to see an integral written that meets such definition). To such purpose i'm trying to construct it by myself however i got stuck...
So can you provide me an example?
Update,
i had a look to the definitions you provided me, i also had a look to the definition the book i'm studying from states. It says that the integral $y = \int_X x d\mu$ is well defined if for each $\Lambda \in X^*$ we have
$$\Lambda y = \int_X \Lambda x d\mu$$
Following this example here and the definition i just gave would be possible to compute something like $$\int_{C^1[0,1]} \left( d/dx \right) d \mu$$ ?
In this case i would suspect the "value" is something like $I$, however how to prove that for each functional $\Lambda$ we have
$$\Lambda I = \int_{C^1[0,1]} \Lambda \left( d/dx \right) d \mu$$
I'm not neither sure of what i'm talking about, but i really would like to understand this concept of "weird integral".
Thank you