Regularization of propagators and logarithms in the background field method in 4-dimensions

The determinant and higher loop terms, usually treated with the Pauli-Villars and higher covariant derivatives methods, in the background field method in 4 dimensions can hardly be regularized simultaneously. At the same time we observe that introduction of a scalar multiplier at the quadratic form, which is equivalent to a change of the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite.