Optimal thresholds for monotone non-Boolean functions

Let left bracket q right bracket equals StartSet 0 comma 1 comma ellipsis comma q minus 1 EndSet [ q ] = { 0 , 1 , … , q − 1 } [q] = \0,1,… ,q-1\ , let normal upper Delta left bracket q right bracket Δ [ q ] Δ [q] denote the simplex of probability measures on left bracket q right bracket [ q ] $[q]$ , and let gamma γ γ denote the Lebesgue measure normalized on normal upper Delta left bracket q right bracket Δ [ q ] Δ [q] . We prove that for any symmetric monotone function f colon left bracket q right bracket Superscript n Baseline right arrow left bracket q right bracket f : [ q ] n → [ q ] 1ptf -1pt [q]ⁿ → [q] and any a element of left bracket q right bracket a ∈ [ q ] a ∈ [q] , we have <jats:alternatives