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Let $\mu_n$ be a sequence of discrete measures on the unit $\ZT=\ZR/\ZZ$ with$\mu_n(0)=0$, and $\mu_n((-\delta,\delta))\to 1$, as $n\to\infty$. We provethat the sequence of convolution operators $(f\ast\mu_n)(x)$ is strong sweepingout, i.e. there exists a set $E\subset\ZT$ such that \md0\lim\sup_{n\to\infty}(\ZI_E\ast\mu_n)(x)= 1,\quad\lim\inf_{n\to\infty}(\ZI_E\ast\mu_n)(x)= 0, \emd almost everywhere on $\ZT$.

On the sweeping out property for convolution operators of discrete measures