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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.
Using a space as a separator does not work if any of the unit codes contains a space. For example, each of the following converts 1.2 km 2 to acres, square yards, and hectares. {{convert|1.2|km2|acre sqyd ha}} → 1.2 square kilometres (300 acres; 1,400,000 sq yd; 120 ha)
PDF does not know of a word space character—the space between two letters and the space between two words differ only in quantity. Therefore, a title with ample letter-spacing for effect will usually end up with spaces in the word processor file, for example INTRODUCTION with spacing of 1 em as I N T R O D U C T I O N on the word processor.
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:
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]
For example Euclidean spaces of dimension n, and more generally n-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the Lebesgue measure. Certain fractal metric spaces such as the Sierpiński gasket can be equipped with the α-dimensional Hausdorff measure where α is the Hausdorff dimension .