Let $\Bbbk$ be a finite-degree field extension of $\mathbb{Q}$. I am trying to figure out the formula for the unique Haar measure $d\mu_{\Bbbk}$ on $\Bbbk$ normalized so that: $$\int_{\Bbbk}\mathbf{1}_{\mathcal{O}_{\Bbbk}}\left(x\right)d\mu_{\Bbbk}\left(x\right)=1$$ where $\mathcal{O}_{\Bbbk}$ is the ring of $\Bbbk$-integers and $\mathbf{1}_{\mathcal{O}_{\Bbbk}}\left(x\right)$ is the indicator function for $\mathcal{O}_{\Bbbk}$.
Since $\Bbbk$ is countable, I'm pretty sure this means that there is some function $f:\Bbbk\rightarrow[0,\infty)$ so that: $$\mu_{\Bbbk}\left(A\right)=\sum_{a\in A}f\left(a\right),\textrm{ }\forall A\subseteq\Bbbk$$ and hence, integration over $\Bbbk$ is really just summation (à la the counting measure). If so, what is the $f$? If not, what is the formula for $\mu_{\Bbbk}\left(A\right)$ for a given $A\subseteq\Bbbk$? I'm interested primarily because I want to try integrating specific functions over $\Bbbk$.