\Psi)$. Then $$\label{eq:transfer1}
\mu(\Lambda_{H}(\psi))=0 ~\forall~\psi \in \Psi  \Rightarrow \mu(\Lambda_{I}(\psi)) = 0 ~\forall~\psi \in \Psi.$$

$(C, \alpha)$-good functions
============================

In this section, we recall the important notion of $(C, \alpha)$-good functions on ultrametric spaces. We follow the treatment of Kleinbock and Tomanov [@KT]. Let $X$ be a metric space, $\mu$ a Borel measure on $X$ and let $(F, |\cdot|)$ be a local field. For a subset $U$ of $X$ and $C, \alpha > 0$, say that a Borel measurable function $f : U \to F$ is $(C, \alpha)$-good on $U$ with respect to $\mu$ if for any open ball $B \subset U$ centred in ${\operatorname{sup}}\mu$ and $\varepsilon > 0$ one has $$\label{gooddef}
\mu \left(\{ x \in B \big| |f(x)| < \varepsilon \} \right) \leq
C\left(\displaystyle \frac{\varepsilon}{\sup_{x \in
B}|f(x)|}\right)^{{\alpha}}|B|,$$ The following elementary properties of $(C,
{\alpha})$-good functions will be used.

1.  If $f$ is $(C,{\alpha})$-good on an open set $V$, so is $\lambda
    f~\forall~\lambda \in
    F$;\

2.  If $f_i, i \in I$ are $(C,{\alpha})$-good on $V$, so is $\sup_{i \in
    I}|f_i|$;\

3.  If $f$ is $(C,{\alpha})$-good on $V$ and for some $c_1,c_2\,\textgreater \,0,\, c_1\leq
    \frac{|f(x)|}{|g(x)|}\leq c_2
    \text{ for all }x \in V$, then g is $(C(c_2/c_1)^{{\alpha}},{\alpha})$-good on $V$.\

4.  If $f$ is 