In Teschl's book on Mathematical Methods in Quantum Mechanics (https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf) in section 4.1 a notion of an integral for Banach space-valued functions is introduced. I would like to know if this notion is known under a specific name and how it relates to other notions of integration, especially the Bochner integral. Furthermore, I would like to know what its advantages are (besides that it is easy to define).
One wants to integrate functions from a compact interval I into a Banach space X.
First, simple functions are defined as usual in the Lebesgue theory (i.e. a function that takes only a finite number of values, where each preimage is Borel measurable). The set of simple functions is denoted $S(I, X)$. The integral for such functions is defined as usual. The completion of $S(I, X)$ w.r.t. the sup-norm is called $R(I, X)$, the space of regulated functions. The integral on this space is defined as the unique continuous extension by means of the B.L.T. theorem. An improper integral over the entire real axis is defined as a principal value (in a way that appears rather ugly to me).
Apparently, this integral notion has some quite nice properties. Especially one can use it to integrate bounded operators and exchange vectors and functionals and multiply operators from the left.
Because the improper integral is defined as a principal value, I assume that this notion of integration is neither weaker nor stronger than the Bochner integral... Am I right? (Analogously to the Riemann vs. Lebesgue situation, although I realize that this notion of integration here is not really analogous to the Riemann integral since it requires measure theory). What about the proper case (on a compact inverval I)?
It seems to me that this is just a generalization of what I would call a Banach space Riemann integral, just using measure theory to make it compatible with spectral calculus...