On the Haar measures in topological fields (Q416447)

The Haar measures \(\mu\) and \(\theta\) for the addition group \((X^+ , +)\) and the multiplication group \((X^{\times} , \times)\) of a locally compact topological field \((X, +, \times)\), respectively, are discussed, and it is shown that the Haar measures \(\mu\) of \((X^+ , +)\) and \(\theta\) of \((X^{\times} , \times)\) are absolutely continuous with respect to each other.NEWLINENEWLINETheorem. The set function \(\theta : S \to \mathbb R\), well defined by \(\theta (E) = \int_E\frac{1}{\varphi(x)} d\,\,\mu (x)\), is the Haar measure on the multiplication group \((X^{\times} , \times)\), where \(S\) is the \(\sigma\)-ring of Borel subsets of \(X\), and the function \(\varphi: X \to \mathbb R\) satisfies:NEWLINENEWLINE(1) \(\forall x\in X \Rightarrow \varphi(x)\geq =; \varphi (x)=0 \Leftrightarrow x=0\);NEWLINENEWLINE(2) \(\forall x, y \in X \Rightarrow \varphi (x \times y) = \varphi(x)\varphi(y)\);NEWLINENEWLINE(3) there exists a positive number \(M\) such that \(\varphi (1_{X^{x}}+a)\leq M\) for \(\forall a \in X\) with \(\varphi (a) \leq 1\), and \(1_{X^{x}}\) the unit of the multiplication group;NEWLINENEWLINE(4) \(\varphi: X \to \mathbb R\) is everywhere continuous.