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Let μ n be a sequence of discrete measures on the unit circle T = R/Z with μ n (0) = 0, and μ n ((−δ, δ)) → 1, as n → ∞. We prove that the sequence of convolution operators (f * μ n )(x) is strong sweeping out; i.e., there exists a set E ⊂ T such that lim sup almost everywhere on T.doi:10.1090/s0002-9939-2010-10829-8 fatcat:qykh6dth2zawdjdfobanqy2f5e