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.

中文翻译：

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Vizing 定理的可测量版本

我们为顶点度和边的多重性分别由 $\Delta$ 和 $\pi$ 统一界定的 Borel 多重图建立了两个版本的 Vizing 定理。“近似”版本指出，对于边集上的任何 Borel 概率测度和任何 $\epsilon>0$，我们可以正确地为除了 $\epsilon $-fraction 的所有边使用 $\Delta+\pi$ 颜色着色以 Borel 的方式。“可测量”版本是我们的主要结果，它指出，如果此外，度量是不变的，那么整个边缘集都有一个可测量的适当边缘着色，最多具有 $\Delta+\pi$ 颜色。