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It is often not possible or desirable to assign a size to ''all'' subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a σ-algebra . Measure theory is that branch of Real Analysis which investigates σ-algebras , measures, Measurable Function s and Integral s. DEFINITION Formally, a measure μ is a Function defined on a σ-algebra Σ over a set ''X'' and taking values in the Extended Interval {Link without Title} such that the following properties are satisfied:
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: The Triple (''X'',Σ,μ) is then called a measure space, and the members of Σ are called '''measurable sets'''. A probability measure is a measure with total measure one (i.e., μ(''X'')=1); a Probability Space is a measure space with a probability measure. For measure spaces that are also Topological Space s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in Analysis (and in many cases also in Probability Theory ) are Radon Measure s. Radon measures have an alternative definition in terms of linear functionals on the Locally Convex Space of Continuous Function s with Compact Support . This approach is taken by Bourbaki(2004) and a number of other authors. For more details see Radon Measure . PROPERTIES Several further properties can be derived from the definition of a countably additive measure. Monotonicity is Monotonic : If and are measurable sets with then . Measures of infinite unions of measurable sets is sequence of sets in , not necessarily disjoint, then :. is continuous from below: If , , , ... are measurable sets and is a subset of for all ''n'', then the Union of the sets is measurable, and :. Measures of infinite intersections of measurable sets is continuous from above: If , , , ... are measurable sets and is a subset of for all ''n'', then the Intersection of the sets is measurable; furthermore, if at least one of the has finite measure, then :. This property is false without the assumption that at least one of the has finite measure. For instance, for each ''n'' ∈ N, let : which all have infinite measure, but the intersection is empty. SIGMA-FINITE MEASURES See Also: Sigma-finite measure A measure space (''X'',Σ,μ) is called finite if μ(''X'') is a finite real number (rather than ∞). It is called ''σ-finite'' if ''X'' can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has ''σ-finite measure'' if it is a union of sets with finite measure. For example, the Real Number s with the standard Lebesgue Measure are σ-finite but not finite. Consider the Closed Interval s {Link without Title} for all Integer s ''k''; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the Real Number s with the Counting Measure , which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to Separability of topological spaces. COMPLETENESS A measurable set ''X'' is called a ''null set'' if μ(''X'')=0. A subset of a null set is called a ''negligible set''. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called ''complete'' if every negligible set is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets ''Y'' which differ by a negligible set from a measurable set ''X'', that is, such that the Symmetric Difference of ''X'' and ''Y'' is contained in a null set. One defines μ(''Y'') to equal μ(''X''). EXAMPLES Some important measures are listed here.
Other measures include: Borel Measure , Jordan Measure , Ergodic Measure , Euler Measure , Gauss Measure , Baire Measure , Radon Measure . NON-MEASURABLE SETS See Also: Non-measurable set Not all subsets of Euclidean Space are Lebesgue Measurable ; examples of such sets include the Vitali Set , and the non-measurable sets postulated by the Hausdorff Paradox and the Banach–Tarski Paradox . GENERALIZATIONS For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a '' Signed Measure '', while such a function with values in the Complex Number s is called a '' Complex Measure ''. Measures that take values in Banach Spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert Space is called a '' Projection-valued Measure ''; these are used mainly in Functional Analysis for the Spectral Theorem . When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term "positive measure" is used. Another generalization is the ''finitely additive measure''. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach Limit s, the dual of ''L''∞ and the Stone-Čech Compactification . All these are linked in one way or another to the Axiom Of Choice . The remarkable result in Integral Geometry known as Hadwiger's Theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on Finite Unions of compact Convex Set s in consists (up to scalar multiples) of one "measure" that is "homogeneous of degree ''k''" for each ''k=0,1,2,...,n'', and linear combinations of those "measures". "Homogeneous of degree ''k''" means that rescaling any set by any factor multiplies the set's "measure" by . The one that is homogeneous of degree ''n'' is the ordinary ''n''-dimensional volume. The one that is homogeneous of degree ''n-1'' is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler Characteristic . SEE ALSO REFERENCES
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