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 $\delta_\gamma \ast \sigma^{\ast m} \ast \exp(\nu)$ for some $\gamma\in \mathbb{R}$, some complex measure $\nu$, some $m\in \mathbb{Z}$ and a given fixed invertible finite signed measure $\sigma$ (which has characteristic function $\mathbb{R} \ni z \mapsto (1+i z)/(1-i z)$). 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 $\delta_\gamma \ast \sigma_1^{\ast m_1} \ast \ldots \ast \sigma_p^{\ast m_p} \ast \exp(\nu)$ for some $\gamma \in \mathbb{R}^n$, some complex measure $\nu$ and $m_1,\ldots, m_p\in \mathbb{Z}$, where the $\sigma_i$ correspond to $\sigma$ 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 $\mathbb{R}^n$.