Let $M = (\mathbb{N}, \mathcal{P}(\mathbb{N}))$ be the D5638: Top subset structure of natural numbers.
Let $|\cdot|$ be the D2184: Set cardinality operation.
The standard natural number counting measure on $M$ is the D4361: Unsigned basic function \begin{equation} \mathcal{P}(\mathbb{N}) \to [0, \infty], \quad E \mapsto |E| \end{equation}