In Borel unbalanced bipartite multigraphs there exists a Borel matching covering μ-almost every vertex in the higher-degree part for any Borel probability measure μ.
Dhawan.Edge-coloring algorithms for bounded degree multigraphs, ACM-SIAM Symposium on Discrete Algorithms (SODA), (2024), 2120–2157
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Measurable matchings in unbalanced graphs
In Borel unbalanced bipartite multigraphs there exists a Borel matching covering μ-almost every vertex in the higher-degree part for any Borel probability measure μ.