  We establish two versions of Vizing's theorem for Borel multi-graphs whose
vertex degrees and edge multiplicities are uniformly bounded by respectively
$\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel
probability measure on the edge set and any $\epsilon>0$, we can properly
colour all but $\epsilon $-fraction of edges with $\Delta+\pi$ colours in a
Borel way. The ``measurable'' version, which is our main result, states that
if, additionally, the measure is invariant, then there is a measurable proper
edge colouring of the whole edge set with at most $\Delta+\pi$ colours.

Combinatorics