Functional integrals are defined in terms of locally compact topological groups and their associated Banach-valued Haar integrals. This approach generalizes the functional integral scheme of Cartier and DeWitt-Morette. The definition allows a construction of functional Mellin transforms. In turn, the functional Mellin transforms can be used to define functional traces, logarithms, and determinants. The associated functional integrals are useful tools for probing function spaces in general and $C^\ast$-algebras in particular. Several interesting aspects are explored.

Expanded and re-posted to the ArXiv as two separate papers titled "A Proposed Definition of Functional Integrals" and "Exploring Functional Mellin Transforms"