The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is local the additive Haar measure $dx$ on $(K,+)$ and the multiplicative Haar measure $d^{*}x$ on $K^{*}$ are related by the equation

\begin{equation*} dx=|x|d^{*}x\text{.} \end{equation*}

So far, so good. Connes now continues by making the claim, which he seems to take for granted as trivial, that if $K$ is global, one has

\begin{equation*} dx=\lim_{\epsilon\longrightarrow 0}\epsilon|x|^{1+\epsilon}d^{*}x\text{.} \end{equation*}

I have no clue why that should be the case. It is very possible that I may not be seeing the forest for all the trees here. Can anyone help?