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Definition

In measure theory, a valuation on a lattice $(L, \leq, \bot, \vee, \top, \wedge)$ is a monotonic function $\mu:L \to [0, \infty]$ with

• a term $\zeta:\mu(\bot) = 0$
• a dependent family of terms
  $$a:L, b:L \vdash m(a, b):\mu(a) + \mu(b) = \mu(a \wedge b) + \mu(a \vee b)$$

  representing the modularity condition.

See also

• probability valuation

• sigma-continuous valuation

References

• Alex Simpson, Measure, randomness and sublocales.