Integration theory for Banach-valued functions I am actually studying integration theory for vector-valued functions in a general Banach space, defining the integral with Riemann's sums.
Everything seems to work exactly as in the finite dimensional case: 
Let X be a Banach space, $f,g \colon I = [a,b] \to X$, $\alpha$, $\beta \in \mathbb{R}$ then:
$\int_I \alpha f + \beta g = \alpha \int_i f + \beta \int_i g$, $\|\int_I f\| \le \int_I \|f\|$, etc...
The fundamental theorem of calculus holds.
If $f_n$ are continuous and uniformly convergent to $f$ it is also true that $\lim_n \int_I f_n = \int_I f$.
My question is: is there any property that hold only in the finite dimensional case? Is it possible to generalize the construction of the integral as Lebesgue did? If so, does it make sense?
Thank you for your help and suggestions
 A: You can define the integral of $f:A \to X$ implicitly by requiring that $\phi( \int_A f) = \int_A \phi (f)$ for every $\phi \in X^\ast$. If it exists, it must be unique by the Hahn-Banach theorem. You may now use your favorite theory for integrating real valued functions to obtain a version for functions with values in a Banach space (or more generally a locally convex TVS). The difficult part is now to find appropriate conditions under which the integral will exist.
Using this implicit definition, we may easily generalize a lot of properties to the infinite dimensional case. An important 
property which does not generalize is that the integral operator $\int_A: \mathcal L^1(A;X) \to X$ will no longer be compact.
A: You might want to have a look to the Bochner-Lebesgue spaces. They are an appropriate generalization to the Banach-space-valued case. Many properties translate directly from the scalar case (Lebesgue theorem of dominated convergence, Lebesgue's differentiation theorem).
Introductions could be found in the rather old book by Yoshida (Functional analysis) or Diestel & Uhl (Vector measures). The latter also considers different (weaker) definitions of integrals.
A: Lebesgue-Bochner integration is well covered in serge lang analysis II -1965 addision wesley or s. lang real and functional analysis -springer verlag 1993.
The theory carries verbatim, but to define integral we can't use supremum of interal of step functions. as there is no order .you have to use limits of generalized step functions. everything goes fine except u may have to replce measurable by mu measurable . evn this will be unnecessary if we define a measurable map is ae limit of simple maps.
another possible aproack is using riemann sums and Henstock-kurzweil integration.an integrable mapping may not be measurable. the absolutely integrable mappings may not form a vector space So L1 space is measurable and absolutely integrable mappings. for details u can refer my notes on net.
A: An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general.
https://en.wikipedia.org/wiki/Bochner_integral
