Full-support probability distribution on a finite alphabet ($\mathrm{supp}(p)=\mathcal{X}$)

A probability mass function $p$ has full support on a specified finite alphabet if $p_x>0$ for every symbol in that alphabet. This is relative to the chosen alphabet: deleting zero-probability symbols produces a smaller alphabet on which the same law has full support. In a weighted prefix code problem, full support makes every shortening of a retained codeword strictly reduce expected codeword length. Without it, a minimum-cost code tree may contain redundant zero-weight branches.