Measure Theory In Topological Vector Spaces
In mathematics, measure theory in topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often infinite-dimensional, but many results of classical measure theory are formulated for finite-dimensional spaces and cannot be directly transferred. This is already evident in the case of the infinite-dimensional Lebesgue measure, Lebesgue measure, which does not exist in general infinite-dimensional spaces. The article considers only topological vector spaces, which also possess the Hausdorff property. Vector spaces without topology are mathematically not that interesting because concepts such as convergence and continuity are not defined there. σ-Algebras Let (X,\mathcal) be a topological vector space, X^* the dual space, algebraic dual space and X' the Dual space#Topological dual space, topological dual space. In topological vector spaces there exist three prominent σ-algebras: * the Borel σ-algebra \mathcal(X): is generate ...
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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 magnitude (mathematics), magnitude, 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, integral, integration theory, and can be generalized to assume signed measure, 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 B ...
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