In this paper, a Lebesgue type theorem on the structure of graphs embedded in the surface of characteristic σ ≤ 0 is given, that generalizes a result of Borodin on plane graphs. As a consequence, it is proved that every such graph without i-circuits for 4 ≤ i ≤ 11 − 3 σ is 3-choosable, that offers a new upper bound to a question of Y. Zhao.