  In this expository paper, we survey nowadays classical tools or criteria used
in problems of convergence everywhere to build counterexamples: the Stein
continuity principle, Bourgain's entropy criteria and Kakutani-Rochlin lemma,
most classical device for these questions in ergodic theory. First, we state a
$L^1$-version of the continuity principle and give an example of its usefulness
by applying it to some famous problem on divergence almost everywhere of
Fourier series. Next we particularly focus on entropy criteria in $L^p$, $2\le
p\le \infty$ and provide detailed proofs. We also study the link between the
associated maximal operators and the canonical Gaussian process on $L^2$. We
further study the corresponding criterion in $L^p$, $1<p<2$ using properties of
$p$-stable processes. Finally we consider Kakutani-Rochlin's lemma, one of the
most frequently used tool in ergodic theory, by stating and proving a criterion
for a.e. divergence of weighted ergodic averages.
