Invertible Complex Measures on Euclidean Spaces

Abstract

In 1971 Taylor characterised all complex measures on $\mathbb{R}$ that are invertible with respect to convolution as those which can be written in the form $\hat{I}{´}_{\hat{I}}^3\ast \ddot{I}{ƒ}^{\ast m}\ast \exp (\hat{I}½)$ for some ${\hat{I}}^3\in \mathbb{R}$, some complex measure $\hat{I}½$, some $m\in \mathbb{Z}$ and a given fixed invertible finite signed measure $\ddot{I}ƒ$ (which has characteristic function $\mathbb{R}\ni z\mapsto (1+iz)/(1-iz)$). We extend Taylor's result to complex measures on ${\mathbb{R}}^n$. Somewhat surprisingly, the structure of invertible complex measures on ${\mathbb{R}}^n$ is not much more complicated than that of complex measures on $\mathbb{R}$, in the sense that they can be represented as $\hat{I}{´}_{\hat{I}}^3\ast \ddot{I}{ƒ}_1^{\ast m_1}\ast \dots \ast \ddot{I}{ƒ}_p^{\ast m_p}\ast \exp (\hat{I}½)$ for some ${\hat{I}}^3\in {\mathbb{R}}^n$, some complex measure $\hat{I}½$ and $m_1,\dots ,m_p\in \mathbb{Z}$, where the $\ddot{I}{ƒ}_i$ correspond to $\ddot{I}ƒ$ in the one-dimensional case and actually live on $1$-dimensional subspaces of ${\mathbb{R}}^n$. Our proof relies on a general result of Taylor for invertible complex measures on locally compact abelian groups. To apply Taylor's result, we extend some existing results for $\mathbb{C}$-valued functions to functions with values in a semisimple commutative unital Banach algebra with connected Gelfand space. The study of invertible complex measures on ${\mathbb{R}}^n$ has some impact on the theory of quasi-infinitely divisible probability distributions on ${\mathbb{R}}^n$.