The source of the following topic is Section 5 (Measure and integration), Chapter II of the book $p$-adic numbers, $p$-adic analysis and zeta functions. by Neal Koblitz:

The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can define Bernoulli distributions by $$\mu_{B,k}(a+p^N \mathbb{Z}_p)=p^{N(k-1)}B_k \left(\frac{a}{p^N}\right), $$ where $B_k(x)$ are Bernoulli polynomials and $B_k=B_k(0)$ are Bernoulli numbers.

These $\mu_{B,k}$ extends to a distribution on $\mathbb{Z}_p$. But these Bernoulli distributions $\mu_{B,k}$ do not define measure on $\mathbb{Z}_p$.

To integrate $p$-adic continuous functions over the compact open subsets $\mathbb{Z}_p$ or $\mathbb{Z}_p^{*}$.of $\mathbb{Q}_p$, one defines the regularized Bernoulli distribution by $$\mu_{k,\alpha}(U)=\mu_{B,k}(U)-\alpha^{-k}\mu_{B,k}(\alpha U)$$ for $\alpha \in \mathbb{Z}_p^{*}$, $k \in \{0\} \cup \mathbb{N}$ and $U=a+p^N \mathbb{Z}_p$.

For $k=1,2, \cdots$, we have the measures $ \mu_{1, \alpha}, \mu_{2,\alpha}, \cdots, \mu_{k, \alpha}, \cdots$. These measure or regularized Bernoulli distributions are related by the following relation $$\int_{\mathbb{Z}_p} f(x) d \mu_{k,\alpha}=\int_{\mathbb{Z}_p} f(x) \cdot kx^{k-1} d \mu_{1, \alpha},$$ for any continuous function $f: \mathbb{Z}_p \to \mathbb{Z}_p$.

This is all the story on $\mathbb{Q}_p$.

Now consider a finite extension $K$ of $\mathbb{Q}_p$. It has similar compact-open subsets $\mathcal{O}_K$, the ring of integers and topological basis consisting of open sets $a+\pi O_K$, where $\pi$ is the uniformizer in the ring $\mathcal{O}_K$.

My question-

Can we extend $\mu_{B,k}$ and $\mu_{k,\alpha}$ from $\mathbb{Z}_p$ to $\mathcal{O}_K$? $$-------------------------------$$ Clearly, we can define Haar measure normalized with $\mu(O_K)=1$ and so $\mu(m_K)=1/q$, where $m_K$ is the maximal ideal and $q=\lvert\mathcal{O}_K/m_K\rvert$.

Also we know that Haar measure $\mu$ coincide with the Bernoulli distribution $\mu_{B,k}$ for $k=0$.

So it appears trivially that we can define the Bernoulli distribution $\mu_{B,0}$ on the finite extension $K$ as it equals to Haar measure and which can be defined on any locally compact Hausdorff space $K$.

I think the only thing we need to handle is the uniformizer $\pi$ which was just $p$ in case of $\mathbb{Z}_p$ in order to extend $\mathcal{O}_K$. If I consider the quadratic extensions $\mathbb{Q}_2(\sqrt{d}), \ d=-1, \pm 3, ...$, then we can choose uniformizer $\pi=2$.

I appreciate any help. Thanks