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

Threshold phenomena on product spaces: BKKKL revisited (once more)