Following recent result of L. M. Tóth [Tót21, Annales Henri Lebesgue, Volume 4 (2021)] we show that every -regular Borel graph with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig’s line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.
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[Kun21] The measurable Hall theorem fails for treeings (2021) (https://arxiv.org/abs/2106.02013)
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[Tót21] Invariant Schreier decorations of unimodular random networks, Ann. Henri Lebesgue, Volume 4 (2021), pp. 1705-1726 | DOI