  In this paper, by establishing a Borel-Cantelli lemma for a capacity which is
not necessarily continuous, and a link between a sequence of independent random
variables under the sub-linear expectation and a sequence of independent random
variables on $\mathbb R^{\infty}$ under a probability, we give the sufficient
and necessary conditions of the strong law of large numbers for independent and
identically distributed random variables under the sub-liner expectation, and
the sufficient and necessary conditions for the convergence of an infinite
series of independent random variables, without any assumption on the
continuity of the capacities. A purely probabilistic proof of a weak law of
large numbers is also given.
  In the version 1, there are errors in the proof of Lemma 2.1 and 2.2. Version
2 corrected the errors under additional conditions, but Corollaries 3.1-3.4 are
only shown for the copy of the random variables in a new sub-linear expectation
space. In this version, we show that Corollaries 3.1-3.4 remain true for the
original random variables and the results for the copy are just special cases.
