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

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.
