  We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) --
referred to as ``BKKKL'' in the title -- about influences on Boolean functions
in order to give a precise statement of threshold phenomenon on the product
space $\{1,..., r\}^\NN$, generalizing one of the main results of a paper by
Talagrand (1994).
