Affine Change-of-Variables Formula in $p$-adic Haar measure integration.

Question

Let $p$ be a prime number, let $V$ be an arbitrary non-empty subset of $\mathbb{Z}_{p}$, and let $d\mu$ be the Haar measure on $\mathbb{Z}_{p}$ subject to the normalization $\int_{\mathbb{Z}_{p}}d\mu=1$. Let $a,b\in\mathbb{Z}_{p}$, with $a\neq0$, and let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ be any $d\mu$-integrable function. Write: $$aV+b=\left\{ ax+b:x\in V\right\}$$ Then, (please answer yes or no) is: $$\int_{V}f\left(ax+b\right)d\mu\left(x\right)=\frac{1}{\left|a\right|_{p}}\int_{aV+b}f\left(x\right)d\mu\left(x\right)$$ the correct change of variables formula (where $\left|\cdot\right|_{p}$ is the $p$-adic absolute value)? If it is not correct, then what is the correct formula?

Answer 1

Sure with $|a|_p = p^{-v_p(a)}$ then $\mu(p^k \Bbb{Z}_p) = p^{-k}$ so $\mu(a^{-1}(p^k \Bbb{Z}_p-b))=|a|_p^{-1} \mu(p^k \Bbb{Z}_p)$, $d\mu(a^{-1}(y-b)) = |a|_p^{-1} d\mu(y)$ and $\int_{V}f\left(ax+b\right)d\mu\left(x\right)=\int_{aV+b}f\left(a(a^{-1}(y-b))+b\right)d\mu\left(a^{-1}(y-b)\right)=\int_{aV+b}f\left(y\right)|a|_p^{-1}d\mu\left(y\right)$

For $f$ continuous the definition is

$$\int_{\Bbb{Z}_p} f(x)d\mu(x) = \lim_{k \to \infty} \sum_{c=0}^{p^k-1}\int_{c+p^k\Bbb{Z}_p} f(x)d\mu(x)\\= \lim_{k \to \infty} \sum_{c=0}^{p^k-1} (f(c)+o(1))\mu(c+p^k \Bbb{Z}_p) = \lim_{k \to \infty} p^{-k}\sum_{c=0}^{p^k-1} f(c)$$