Convolution of Vector Measures on Locally Compact Groups

We establish two definitions of the convolution of vector measures on locally compact groups by employing injective tensor integration. These two formulations are shown to be isomorphic. We further investigate fundamental properties of the convolution of vector measures, including a representation in terms of double integrals and its behavior under the Fourier transform. In particular, we demonstrate that the Fourier transform of the convolution admits a factorization analogous to the classical case, with an inherent asymmetry arising from the vector-valued setting.