Fine path maps and public history maps

Let a leaf public history be $a=(a_1,\dots ,a_m)$, with path nodes $v_0,\dots ,v_{m-1}$. For its Kraus refinement $\mu =({\mu }_1,\dots ,{\mu }_m)$, define

Each product is between its stated successive spaces. The fine index $(a,\mu )$ is not necessarily a list of physically observed outcomes.