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An affine utility on probability measures functional satisfies $U_0(p\lambda +(1-p)\mu )=p{U}_0(\lambda )+(1-p)U_0(\mu )$ on mixtures of probability measures. Strict preferences are represented by comparing its real values, and indifference means equal values. This is cardinal utility on lotteries rather than merely an ordinal ordering of deterministic outcomes.

Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.

An affine function preserves mixtures, soThereforeThe positive mixture weight preserves the strict sign; the common component cancels. This is the independence axiom for lottery preferences for the affine utility on probability measures.