  In this paper, we establish some general forms of the law of the iterated
logarithm for independent random variables in a sub-linear expectation space,
where the random variables are not necessarily identically distributed.
Exponential inequalities for the maximum sum of independent random variables
and Kolmogorov's converse exponential inequalities are established as tools for
showing the law of the iterated logarithm. As an application, the sufficient
and necessary conditions of the law of iterated logarithm for independent and
identically distributed random variables under the sub-linear expectation are
obtained. In the paper, it is also shown that if the sub-linear expectation
space is rich and regular enough, it will have no continuous capacity. The laws
of the iterated logarithm are established without the assumption on the
continuity of capacities.
  This revision fills a gap in the proof of Proposition 4.2 of arXiv:2103.01390
under an additional assumption.
