Valuation (measure Theory)
In measure theory, or at least in the approach to it via the domain theory, a valuation is a Map (mathematics), map from the class of open sets of a topological space to the set of positive number, positive real numbers including infinity, with certain properties. It is a concept closely related to that of a Measure (mathematics), measure, and as such, it finds applications in measure theory, probability theory, and theoretical computer science. Domain/Measure theory definition Let \scriptstyle (X,\mathcal) be a topological space: a valuation is any set function v : \mathcal \to \R^+ \cup \ satisfying the following three properties \begin v(\varnothing) = 0 & & \scriptstyle\\ v(U)\leq v(V) & \mbox~U\subseteq V\quad U,V\in\mathcal & \scriptstyle\\ v(U\cup V)+ v(U\cap V) = v(U)+v(V) & \forall U,V\in\mathcal & \scriptstyle\, \end The definition immediately shows the relationship between a valuation and a measure: the properties of the two mathematical object are often very similar i ...
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Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, ...
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