The mathematics of covariant spectral regularization for covariant matrix valued measures with one Lorentz index on open subsets of Minkowski space is described. This is then applied to the case of the vertex function and an expression for the density associated with the vertex function in the t channel with respect to Lebesgue measure on Minkowski space is obtained. This density is well defined, non-divergent and analytic over its domain of definition and is obtained without using renormalization or needing to consider final state radiation. The limit of the expression for the vertex function in the t channel at low energy and low momenta is computed resulting in the classical result for the leading order (LO) contribution to the anomalous magnetic moment of the electron.