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Following recent result of L. M. T\' oth [arXiv:1906.03137] we show thatevery $2\Delta$-regular Borel graph $\mathcal{G}$ with a (not necessarilyinvariant) Borel probability measure admits approximate Schreier decoration. Infact, we show that both ingredients from the analogous statements for finitegraphs have approximate counterparts in the measurable setting, i.e.,approximate K\H{o}nig's line coloring Theorem for Borel graphs without oddcycles and approximate balanced orientation for even degree Borel graphs.

Approximate Schreier decorations and approximate K\H{o}nig's line coloring Theorem