Remark 17.3.1. In the proof of Singer’s Representation Theorem, the $E^{*}$-valued measure is constructed pointwise as
\[\nu: \cb_{X} \to E^{*} \quad \dpn{y,\nu(A)}{E}= \int_{X \times B}\one_{A}(x) \cdot \dpn{y, \phi}{E}\mu(dx, d\phi)\]
It may be tempting to use the strong formulation directly
\[\nu: \cb_{X} \to E^{*} \quad \nu(A) = \int_{X \times B}\phi \cdot \one_{A}(x) \mu(dx, d\phi)\]
However, without additional assumptions on $E^{*}$, $\phi \cdot \one_{A}(x)$ may not be strongly measurable, which prevents this direct use of the Bochner integral. Thus the weak formulation is a necessary complication in the proof.