  The famous List Colouring Conjecture from the 1970s states that for every
graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In
1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds
asymptotically. Our main result is a local generalization of Kahn's theorem.
More precisely, we show that, for a graph $G$ with sufficiently large maximum
degree $\Delta$ and minimum degree $\delta \geq \ln^{25} \Delta$, the following
holds: for every assignment of lists of colours to the edges of $G$, such that
$|L(e)| \geq (1+o(1)) \cdot \max\left\{\rm{deg}(u),\rm{deg}(v)\right\}$ for
each edge $e=uv$, there is an $L$-edge-colouring of $G$. Furthermore, Kahn
showed that the List Colouring Conjecture holds asymptotically for linear,
$k$-uniform hypergraphs, and recently Molloy generalized Kahn's original result
to correspondence colouring as well as its hypergraph generalization. We prove
local versions of all of these generalizations by showing a weighted version
that simultaneously implies all of our results.
