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

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.
