Abstract: In 1982 Y. Katznelson and B. Weiss gave new and simple proofs of Birkhoff's and Kingman's ergodic theorems, using an elegant stopping time argument. This idea was later applied by A. Furman and then by S. Jitomirskaya and R. Mavi to the study of the maximal Lyapunov exponent of a sub-additive cocycle, leading to a proof of its uniform (in phase and cocycle) upper semi-continuity. Their result, however, requires unique ergodicity, a property that systems like Bernoulli shifts do not satisfy. Compromising on succinctness (which this abstract also does) and on elegance, we proved a weaker but more general version of this result, one which applies to Bernoulli shifts as well. In turn, this became a tiny but essential wheel in a mechanism P. Duarte and I have developed to prove continuity of Lyapunov exponents of linear cocycles in a general setting. The talk will not compromise on the elegance of the original argument. [Based on joint work with Pedro Duarte.]