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

For a probability measure $\mu$ on a topological space, its support isA point belongs to the support when observations can fall arbitrarily close to it with positive probability. This is a neighbourhood condition, not a requirement that the point itself have positive probability mass or that a chosen density be positive there.

For the uniform distribution on $(0,1)$ viewed as a law on the real line, the support is $[0,1]$, including both endpoints. A density $f(x)=3x^2/2$ on $[-1,1]$ vanishes at zero, but zero still lies in its support because every open neighbourhood of zero has a positive integral of $f$. A discrete distribution with mass $2^{-n}$ at $1/n$, $n\ge 1$, likewise has zero in its support despite assigning it probability mass zero.

If a distribution's support equals an interval, every open subinterval of that interval has positive probability. The weaker condition $\mu (I)=1$ does not imply that the support equals $I$: a uniform law on $(0,1)$ also assigns probability one to $[0,2]$, but assigns zero to its subinterval $(3/2,7/4)$. This distinction matters when using support to describe feasible observations or randomized effort in a contest.

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.