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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 metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. [b] The least such r is called the diameter of M. The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded.
Units that measure reciprocal dimensions can be converted (e.g., S to megohm). Parentheses for grouping are supported. This sometimes allows more natural expressions, such as in the example given in Complex units expressions. Roots of units (e.g., sqrt((lbf/inch) / lb) can be computed. Nonlinear units conversions (e.g., °F to °C) are supported.
In measure theory, projection maps often appear when working with product (Cartesian) spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable.
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 measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume.
The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not ...
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm. Then ba(Σ) = B(Σ)* is the continuous dual space of B(Σ). This is due to Hildebrandt [4] and Fichtenholtz & Kantorovich. [5] This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable ...