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Definition

In measure theory, a $\sigma$-continuous probability valuation on a $\sigma$-complete lattice $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a probability valuation $\mu:L \to [0, 1]$ with

• a family of dependent terms
  $$s:\mathbb{N} \to L \vdash c(s):\left(\prod_{n:\mathbb{N}} s(n) \leq s(n + 1)\right) \to \left(\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n)) \right)$$

  representing the $\sigma$-continuity condition.

See also

• unit interval

• sigma-continuous valuation

• probability valuation

• probability measure

References

• Alex Simpson, Measure, randomness and sublocales.