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The two‐weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures   σ   and   w   on   R  . In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two‐weight problem by Lacey, Sawyer, Shen, and Uriarte‐Tuero. Our characterization is in terms of Sawyer‐type testing conditions and a variant of the two‐weight    A  2    condition, where   σ   and   w   are integrated over complementary intervals only. A key novelty of the proof is a two‐weight inequality for the Poisson integral with ‘holes’.

The two‐weight inequality for the Hilbert transform with general measures