Wikipedia http://en.wikipedia.org/wiki/Pettis_integral defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality.
It calls the Pettis & Bochner integral the weak & strong integral respectively, which implies some kind of relationship; also, apparently there is a Dunford integral which specialises to the Pettis integral.
My question is: Why are these weak integrals useful, what is the definition of the Dunford Integral and how does it specialise to the Pettis one, and what is the relationship between the Pettis Integral & the Bochner Integral?.
I've just noted: Weak and Strong Integration of vector-valued functions, which answers the part of my question about the Pettis Integral.
Best Answer
Let $f \colon \Omega \to E$ be your function. $\mu$ is a measure on $\Omega$. Assume, for every $x^* \in E^*$, the composition $x^* \circ f$ is $\mu$-integrable. The Dunford integral in general lies in $E^{**}$, namely $\int_A f d\mu$ is the element $u^{**} \in E^{**}$ defined by $u^{**}(x^*) = \int_A x^{*}\circ f\,d\mu$ for all $x^* \in E^*$. And then, of course, if $\int_A f\,d\mu \in E$ for all $A$, we say $f$ is Pettis integrable.
If $f$ is Bochner integrable, then it is also Pettis integrable, and the two integrals agree.
If $f$ has almost all its values in a separable space, then $f$ is Bochner integrable if and only if it is Pettis integrable, and the scalar integral $\int \|f\| d\mu$ converges.