  We prove a full measurable version of Vizing's theorem for bounded degree
Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree
uniformly bounded by $\Delta\in \mathbb{N}$ defined on a standard probability
space $(X,\mu)$ admits a $\mu$-measurable proper edge coloring with
$(\Delta+1)$-many colors. This answers a question of Marks [Question 4.9, J.
Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of
[Problem 6.13, survey (2020)], and extends the result of the author and
Pikhurko [Adv. Math. 374, (2020)] who derived the same conclusion under the
additional assumption that the measure $\mu$ is $\mathcal{G}$-invariant.
