  Let X_1,..., X_n be independent Bernoulli random variables and $f$ a function
on {0,1}^n. In the well-known paper (Talagrand1994) Talagrand gave an upper
bound for the variance of f in terms of the individual influences of the X_i's.
This bound turned out to be very useful, for instance in percolation theory and
related fields. In many situations a similar bound was needed for random
variables taking more than two values. Generalizations of this type have indeed
been obtained in the literature (see e.g. (Cordero-Erausquin2011), but the
proofs are quite different from that in (Talagrand1994). This might raise the
impression that Talagrand's original method is not sufficiently robust to
obtain such generalizations. However, our paper gives an almost self-contained
proof of the above mentioned generalization, by modifying step-by-step
Talagrand's original proof.
