AbstractNecessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {einx;x∈R}∞n=−∞ or {einx;x∈R}∞n=1 in Lα(μ, R), α⩾1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {einx;x∈R}∞n=−∞ is dense in Lα(μ, R) α⩾1. The converse statement is also true.