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Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals

$$\int\limits_{k^n} f(x_1, ... , x_n) d\mu_{\operatorname{Haar}}(x)$$

for suitable analytic $k$-valued functions $f$ on $k^n$. If $\varphi: U \rightarrow V$ is a $k$-analytic isomorphism between open sets $U$ and $V$ in $k^n$, then for suitable complex valued functions $f: k^n \rightarrow \mathbb C$, we have a change of variables formula

$$\int\limits_V f(x)dx_1 \cdots dx_n = \int\limits_U f \circ \varphi(x) |D_{\varphi}(x)|dx_1 \cdots dx_n$$ where $D_{\varphi}$ is the determinant of the Jacobian of $\varphi$, and $dx_1 \cdots dx_n$ is the (real valued) Haar measure on $k^n$. Is there an analogue of this formula for $k$-valued measures and $k$-analytic maps $f: k^n \rightarrow k$? Something like

$$\int\limits_V f(x) d\mu_{\operatorname{Haar}}(x) = \int\limits_U f \circ \varphi(x) D_{\varphi}(x) d\mu_{\operatorname{Haar}}(x)$$ (without the absolute value)?

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    $\begingroup$ What kind of "p-adic Haar measure" are you using here? Obviously whether or not such a thing exists depends on what properties that you want it to have, but for most reasonable definitions, such a thing doesn't exist. $\endgroup$ Jan 24, 2020 at 7:24
  • $\begingroup$ I don't really know. The motivation behind this is I'm repeatedly encountering two $p$-adic Jacobian determinants whose absolute values are equal (which I determine via the usual complex change of variables formula). I want to say that they are actually equal, so I was hoping I could see this by some $p$-adic valued integration. $\endgroup$
    – D_S
    Jan 24, 2020 at 15:06
  • $\begingroup$ So you "don't really know" how to justify the assertion you make in the first sentence of your own question? $\endgroup$ Jan 24, 2020 at 15:58
  • $\begingroup$ No I don't $\space$ $\endgroup$
    – D_S
    Jan 24, 2020 at 22:45
  • $\begingroup$ @David Loeffler: Maybe a more interesting question would be whether there is some sort of p-adic measure / distribution / something for which the stated change of variables formula holds, at least for some sufficiently nice class of functions? $\endgroup$
    – DCM
    Jan 27, 2020 at 10:24

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