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The popularity of Minecraft mods has been credited for helping Minecraft become one of the best-selling video games of all time. The first Minecraft mods worked by decompiling and modifying the Java source code of the game. The original version of the game, now called Minecraft: Java Edition, is still modded this way, but with more advanced tools.
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.
In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces.In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts," using the counting measure on some discrete space. [1]
Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure or distribution. See the list of probability distributions for instances.
An example of a measure on the real line with its usual topology that is not outer regular is the measure where () =, ({}) =, and () = for any other set .; The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure.
i.e. the space of finitely additive signed measures on Σ that are absolutely continuous with respect to μ (μ-a.c. for short). When the measure space is furthermore sigma-finite then L ∞ (μ) is in turn dual to L 1 (μ), which by the Radon–Nikodym theorem is identified with the set of all countably additive μ-a.c. measures. In other ...
Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ 0, μ 0) of this measure space that is complete. [3] The smallest such extension (i.e. the smallest σ-algebra Σ 0) is called the completion of the measure space. The completion can be constructed as follows:
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in L p in terms of convergence in measure and a condition related to uniform integrability.