Definition 29.1.2 (Weak Integrability*).label Let $(X, \cm, \mu)$ be a measure space, $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$, and $f: X \to E$, then $f$ is Dunford $\lambda$-integrable* if:

1. (I1)

   $f$ is weakly measurable.

2. (I2)

   For each $\phi \in F$, $\phi \circ f \in L^{1}(X; K)$.

3. (I3)

   For each $A \in \cm$, the mapping

   \[F \to K \quad \phi \mapsto \int_{A} \dpn{f(x), \phi}{\lambda}d\mu\]

   
   

   is a continuous linear functional on $F$.

For each $A \in \cm$, the element $\phi \mapsto \int_{A} \dpn{f(x), \phi}{\lambda}d\mu$ of $F^{*}$ is the Dunford $\lambda$-integral of $f$ over $A$, denoted $\int_{A}^{*} f d\mu$.

The function $f$ is Pettis $\lambda$-integrable* if it satisfies (I1), (I2), and

1. (I3+)

   For each $A \in \cm$, the mapping

   \[F \to K \quad \phi \mapsto \int_{A} \dpn{f(x), \phi}{\lambda}d\mu\]

   
   

   is a $\sigma(F, E)$-continuous linear functional on $F$.

In which case, for each $A \in \cm$, $\int_{A}^{*} f d\mu$ is the Pettis $\lambda$-integral of $f$ over $A$.