Marks proved that the analogue of Vizing's Theorem in the context of bounded degree Borel graphs does not hold, i.e., if D is the maximum degree of a Borel graph G, then the Borel chromatic index (Borel edge chromatic number) can be strictly bigger than D+1. I will discuss the situation of Vizing's Theorem for bounded degree Borel graphs with probability measure (the measurable chromatic index), especially the case when the measure in question is invariant. This is a joint work with Oleg Pikhurko.