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

Following recent result of L. M. T\' oth [arXiv:1906.03137] we show that every $2\Delta$-regular Borel graph $\mathcal{G}$ 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 K\H{o}nig's line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.