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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.
While MS-DOS and NT always treat the suffix after the last period in a file's name as its extension, in UNIX-like systems, the final period does not necessarily mean that the text after the last period is the file's extension. [1] Some file formats, such as .txt or .text, may be listed multiple times.
The term Borel space is used for different types of measurable spaces. It can refer to any measurable space, so it is a synonym for a measurable space as defined above [1] a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel -algebra) [3]
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:
A simple example is a volume (how big an object occupies a space) as a measure. 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, mass, and probability of events. These seemingly distinct concepts have many similarities and ...
an Office suite; allows to export (and import, with accuracy limitations) PDF files. Microsoft Word 2013: Proprietary: Desktop software. The 2013 edition of Office allows PDF files to be converted into a format that can be edited. Nitro PDF Reader: Trialware: Text highlighting, draw lines and measure distances in PDF files. Nitro PDF Pro ...
The classical definition of a locally integrable function involves only measure theoretic and topological [4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ): [5] however, since the most common application of such functions is to distribution theory on Euclidean spaces, [2] all ...
The space of all countable ordinals with the topology generated by "open intervals" is a locally compact Hausdorff space. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.