Knowra Measure space Measure space A measure space is a set equipped with a sigma-algebra of measurable subsets and a measure assigning each such subset a nonnegative size, possibly infinite.
Set : A collection of distinct objects treated as a single mathematical entity. The underlying points of a measure space form a set.
Nonnegative measure : A measure whose values lie between zero and positive infinity, including both endpoints. Ordinary measure spaces use nonnegative sizes for measurable subsets.
Lebesgue integral : An integral defined using measurable functions and measures, built by approximation with simple functions. It integrates functions on measure spaces beyond the limits of elementary integration.
Lebesgue measure : The standard translation-invariant measure on Euclidean spaces, extending ordinary length, area, and volume. It assigns familiar geometric sizes within a measure-space framework.
Outer measure : A countably subadditive size function defined on all subsets of a set, often used to construct a measure. Unlike a measure on a sigma-algebra, it assigns values before measurability is restricted.
Sigma-algebra : A collection of subsets containing the whole set and closed under complements and countable unions. It specifies exactly which subsets can receive a measure.
Null set : A measurable set assigned measure zero, though it may contain points. Zero size does not imply emptiness in a measure space.
Probability space : A measure space whose measure of the entire sample space is one. Probability theory is measure theory with total measure normalized to one.
Counting measure : A measure that assigns each set its number of elements, with infinite value for infinite sets. It makes discrete counting a direct example of measure.
Premeasure : A countably additive set function defined on an algebra of sets, intended for extension to a sigma-algebra. It supplies partial size data before a full measure space is formed.
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