<abstract><p>The aim of this paper is to study and establish precise asymptotics for complete integral convergence in the law of the logarithm under the sub-linear expectation space. The methods and tools in this paper are different from those used to study precise asymptotics theorems in probability space. We extend precise asymptotics for complete integral convergence from the classical probability space to sub-linear expectation space. Our results generalize corresponding results obtained by Fu and Yang^{[<xref ref-type="bibr" rid="b13">13</xref>]}. We further extend the limit theorems in classical probability space.</p></abstract>