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In this paper, by establishing a Borel-Cantelli lemma for a capacity which isnot necessarily continuous, and a link between a sequence of independent randomvariables under the sub-linear expectation and a sequence of independent randomvariables on $\mathbb R^{\infty}$ under a probability, we give the sufficientand necessary conditions of the strong law of large numbers for independent andidentically distributed random variables under the sub-liner expectation, andthe sufficient and necessary conditions for the convergence of an infiniteseries of independent random variables, without any assumption on thecontinuity of the capacities. A purely probabilistic proof of a weak law oflarge numbers is also given.  In the version 1, there are errors in the proof of Lemma 2.1 and 2.2. Version2 corrected the errors under additional conditions, but Corollaries 3.1-3.4 areonly shown for the copy of the random variables in a new sub-linear expectationspace. In this version, we show that Corollaries 3.1-3.4 remain true for theoriginal random variables and the results for the copy are just special cases.

The sufficient and necessary conditions of the strong law of large numbers under the sub-linear expectations