Lebesgue integral with respect to vector measures?

Question

Good evening,

I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense of Riemann, not of Lebesgue. And the proofs for Riemann integrals are often long.

Secondly, in the book of Rudin, Functional Analysis, the author doesn't define an integral of Lebesgue's type with respect to the spectral measure. In stead, he always wants the readers to understand the integral with respect to the spectral measure as in the scalar case. Precisely, let $T$ be a normal operator on a hilbert space $H,$ and let $T = \int_{z\in\sigma(T)} z dE(z)$ be the spectral decomposition of $T.$ The integral has to be understood as $\langle Tx,y\rangle = \int_{\sigma(T)}z dE_{x,y}(z)$, where $E_{x,y}$ is the scalar measure defined by $E_{x,y}(\omega) = \langle E(\omega)x,y\rangle$ for all $\omega$ borelian sets of $\sigma(T)$, $x, y \in H$ and $\langle \cdot,\cdot\rangle$ the inner product of $H.$

My questions : Can these integrals be understood in the sense of Lebesgue? What is a good introductory reference for the theory of Lebesgue integral with respect to vector measures (of course, if this theory exists)? What are difficulties when we construct such theory?

Maybe, my questions are not well written, because of my limited english knowledge. I hope you understand my post. Any help is appreciated.

Thanks in advance,

Duc Anh

Answer 1

Marc Rieffel has some notes that develop integration with respect to Banach-space valued measures from the ground up. The notes are very thorough. They are available here:

http://math.berkeley.edu/~rieffel/measinteg.html

Lectures notes from 1970 for the first-year graduate-level analysis course on measures and integration at UC Berkeley that I gave several times during the late 1960's can be found here. The notable feature of the notes is that they treat the Bochner integral from the beginning, in a quite elementary way (e.g. no mention of the Hahn-Banach theorem). This has both practical and pedagogical advantages. Not all lectures listed in the table of contents were ever typed up.

The origin of these lecture notes lies in the turmoil on the UC Berkeley campus in the late 1960's, when there were periods of time when students indicated that as a protest they did not want to come to class (and if they did try to come to class there was a significant probability that they would encounter tear-gas or worse), but they indicated that they wanted to continue their studies and so requested that written notes of the lectures they missed (if held at all) be made available to them.

Answer 2

I didn't know the reference above :) So far, the last survey about it was due to J. Diestel and J. Uhl, Vector Measures. By the way no pdf. available.