Let \(\mu\) be a measure on \({\mathbb{R}}_+^{n+1}\) and \(\omega\) a nonnegative weight in \({\mathbb{R}}^ n\). In this paper, the author gives a condition on the pair (\(\mu\),\(\omega)\), under which the Poisson integral is a bounded operator from \(L^ p({\mathbb{R}}^ n;\omega (x)dx)\) into \(weak\)-L\({}^ p({\mathbb{R}}_+^{n+1},\mu)\). As a corollary he finds some results of Carleson, Fefferman-Stein and Muckenhoupt.