Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

中文翻译：

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普遍可测同态的连续性

回答克里斯滕森关于 Haar 空集的开创性工作中长期存在的问题 [数学。扫描。 28(1971), 124–128;以色列 J. 数学。 13(1972), 255–260;拓扑和Borel 结构。描述性拓扑和集合论及其在泛函分析和测度论中的应用, 北荷兰数学研究, 10 (Notas de Matematica, No. 51)。(North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp]，我们表明波兰群之间普遍可测量的同态是自动连续的。使用我们对群同态连续性的一般分析，该结果用于校准波兰群之间不连续同态存在的强度。特别是，它表明，模 $\text{ZF}+\text{DC}$ ，波兰群之间不连续同态的存在意味着汉明图 $\{0,1\}^{\mathbb{N}}$ 有有限的色数。