On continuity of the variation and the Fourier transform

Abstract

Let $S$ be a commutative semitopological semigroup with identity and involution, $\Gamma$ a compact subset in the topology of pointwise convergence of the set of semicharacters on $S$. Let $f$ be a function which admits a (necessarily unique) integral representation of the form $$f(s)=\int_{\Gamma}{\rho(s)d\mu_{f}(\rho)}\quad (\rho \in \Gamma,s \in S$$ with respect to a complex regular Borel measure $\mu_{f}$ on $\Gamma$. The function $|f|(\cdot)$ defined by $|f|(s)=\int_{\Gamma}{\rho(s)d|\mu_{f}|}$ is called the variation of $f$. It is shown that the variation $|f|$ is bounded and continuous if and only if $f$ is also bounded and continuous. This, coupled with the author's previous characterization of functions of bounded variation, gives a new description of the Fourier transforms of bounded measures on locally compact commutative groups.