Nonstandard representations of generalized sections of vector bundles (Q1911596)

One proves that every generalized section \(T\) of a \(C^\infty\)-vector bundle \(E\) over a \(\sigma\)-compact manifold \(M\) can be represented by a \(*\)-integral in the sense that there exists a \(^*C^\infty\)-internal section \(\beta_T\) of the nonstandard extension \(^*E\) of \(E\) such that \[ T(u) = \int_{^*M} \beta_T \cdot ^*u \] for every compactly supported \(C^\infty\)-section \(u\) of \(E^t \otimes |\Lambda_M|,\) where \(E^t\) is the dual bundle of \(E\) and \(|\Lambda_M |\) stands for the density bundle over \(M\), see also [\textit{T. Kawai}, Stud. Logic Found. Math. 111, 55-76 (1983; Zbl 0542.03046)].