Support of a probability distribution ($\mathrm{supp}\mu =\{x:\mu (U)>0\text{ for every open }U\ni x\}$)

For a probability measure on a topological space, its support consists of the points whose every open neighborhood has positive probability. For a continuous effort distribution supported on an interval, every open subinterval of that interval has positive probability.

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.

An effort support is the support of a probability distribution used to randomize effort in a contest. The all-pay indifference equation with random entry determines this support together with its distribution function.