  The classical theorem of Vizing states that every graph of maximum degree $d$
admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was
earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite.
  We investigate the existence of measurable edge-colorings for graphings. A
graphing is an analytic generalization of a bounded-degree graph that appears
in various areas, such as sparse graph limits, orbit equivalence theory and
measurable group theory. We show that every graphing of maximum degree $d$
admits a measurable edge-coloring with $d + O(\sqrt{d})$ colors; furthermore,
if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a
certain conjecture about finite graphs that strengthens Vizing's theorem is
true, then our method will show that $d+1$ colors are always enough.
