nLab sigma-continuous probability valuation

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Definition

In measure theory, a σ\sigma-continuous probability valuation on a σ \sigma -complete lattice (L,,,,,,)(L, \leq, \bot, \vee, \top, \wedge, \Vee) is a probability valuation μ:L[0,1]\mu:L \to [0, 1] such that the σ\sigma-continuity condition is satisfied: for all sequences s:Ls:\mathbb{N} \to L, if s(n)s(n+1)s(n) \leq s(n + 1) for all natural numbers nn \in\mathbb{N}, then

μ( n:s(n))sup n:μ(s(n))\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n))

See also

References

Last revised on October 25, 2023 at 01:34:09. See the history of this page for a list of all contributions to it.