Gleason's theorem and Cauchy's functional equation (Q2365477)

The author describes regular and bounded measures on the effect algebra of the closed interval \([0,1]\) (\(a,b \in [0,1]\) are orthogonal iff \(a+b \leq 1\), in this case \(a \oplus b = a+b\) is defined) and shows that every bounded measure is a multiple of the identity. (The Gleason theorem is used.) This gives a solution of Cauchy's functional equation \(f(x+y) = f(x)+f(y)\) for \(x,y,x+y \in [0,1]\).