Abstract
We study measures on the effect algebras of the closed interval [0, 1] and we describe regular or bounded measures. Applying Gleason's theorem for measures on the system of all closed subspaces of a Hilbert space, we show that any bounded measure m has the formm(t)=tm 1, t ∈ [0, 1], and as a by-product it gives a solution of Cauchy's basic functional equationf (x+ y)=f(x) + f(y) forx, y,x + y ∈ [0, 1].
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Dvurečenskij, A. Gleason's theorem and Cauchy's functional equation. Int J Theor Phys 35, 2687–2695 (1996). https://doi.org/10.1007/BF02085773
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DOI: https://doi.org/10.1007/BF02085773