Measurable versions of Vizing's theorem

We establish two versions of Vizing’s theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\mathrm{\Delta }$ 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 $\mathrm{\Delta }+\pi$ colours.