# Lebesgue integral with respect to vector measures?

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.

Duc Anh