  In 1948 Feynman introduced functional integration. Long ago the problematic
aspect of measures in the space of fields was overcome with the introduction of
volume elements in Probability Space, leading to stochastic formulations. More
recently Cartier and DeWitt-Morette focused on the definition of a proper
integration measure and established a rigorous mathematical formulation of
functional integration. Their central observation relates to the distributional
nature of fields, for it leads to the identification of distribution
functionals with Schwartz space test functions as density measures. This is
just the mathematical content of the Taylor-Lagrange Scheme developed by the
authors in a recent past. In this scheme fields are living in metric
Schwartz-Sobolev spaces, subject to open coverings with subordinate
partition-of-unity test functions. In effect these PU, through the convolution
operation, lead to smooth field functions on an extended Schwartz space. In
this way the basic assumption of differential geometry -- that fields live on
differentiable manifolds -- is validated. Next it is shown that convolution in
the theory of distributions leads to a sound definition of Laplace-Stieltjes
transforms , stemming from the existence of an isometry invariant Hausdorff
measure in the space of fields. Turning to gauge theories the construction of
smooth vector fields on a curved manifold is established, as required for
differentiable vector fibre bundles. The proper choice of a connection further
separates in the functional Hausdorff integration measure a finite Gaussian
integration over the gauge parameter which factors out and plays no physical
role. For non-Abelian Yang-Mills gauge theories the properties of
Vilkowisky-DeWitt's connection and of Landau-DeWitt's covariant back-ground
gauge result in very simple calculation rules.
