Convex measure: Difference between revisions

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==General definition and special cases==

Let ''X'' be a [[locally convex space|locally convex]] [[Hausdorff space|Hausdorff]] [[vector space]], and consider a probability measure ''μ'' on the [[Borel sigma-algebra|Borel ''σ''-algebra]] of ''X''. Fix a −∞ ≤ ''s'' ≤ 0, and define, for ''u'', ''v'' ≥ 0 and 0 ≤ ''λ'' ≤ 1,

:<math>M_{s, \lambda}(u, v) = \begin{cases} (\lambda u^s + (1 - \lambda) v^{s})^{1/s} & \text{if } - \infty < s < 0, \\ \min(u, v) & \text{if } s = - \infty, \\ u^{\lambda} v^{1- \lambda} & \text{if } s = 0. \end{cases}</math>

For subsets ''A'' and ''B'' of ''X'', we write