σ-finite Measure
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, given a positive or a signed measure \mu on a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
(X, \mathcal F), a \sigma-finite subset is a measurable subset which is the union of a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
number of measurable subsets of finite measure. The measure \mu is called a \sigma-finite measure if the set X is \sigma-finite. A finite measure, for instance a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
, is always \sigma-finite. A different but related notion that should not be confused with \sigma-finiteness is s-finiteness.


Definition

Let (X, \mathcal) be a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
and \mu a measure on it. The measure \mu is called a σ-finite measure, if it satisfies one of the four following equivalent criteria: # the set X can be covered with at most countably many
measurable set 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 hav ...
s with finite measure. This means that there are sets A_1, A_2, \ldots \in \mathcal A with \mu\left(A_n\right) < \infty for all n \in \N that satisfy \bigcup_ A_n = X . # the set X can be covered with at most countably many measurable disjoint sets with finite measure. This means that there are sets B_1, B_2, \ldots \in \mathcal A with \mu\left(B_n\right)< \infty for all n \in \N and B_i \cap B_j = \varnothing for i \neq j that satisfy \bigcup_ B_n = X. # the set X can be covered with a monotone sequence of measurable sets with finite measure. This means that there are sets C_1, C_2, \ldots \in \mathcal with C_1 \subset C_2 \subset \cdots and \mu\left(C_n\right) < \infty for all n \in \N that satisfy \bigcup_ C_n = X . # there exists a strictly positive
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
f whose integral is finite. This means that f(x) > 0 for all x \in X and \int f(x) \mu(\mathrmx)<\infty. If \mu is a \sigma-finite measure, the
measure space 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 -algebra) and the method that ...
(X, \mathcal, \mu) is called a \sigma -finite measure space.


Examples


Probability measure

If (X, \mathcal, \mu) is a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
, then the probability measure, \mu is σ-finite, because X is trivially covered by itself: \mu(X)=1.


Lebesgue measure

For example,
Lebesgue measure 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 '-spaces. For lower dimensions or , it c ...
on the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is not finite, but it is σ-finite. Indeed, consider the intervals for all
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s ; there are countably many such intervals, each has measure 1, and their union is the entire real line. For Lebesgue measure on \R^n a similar disjoint cover can be constructed using unit-volume ''n''-cubes (criterion 2); or by a monotone sequence of expanding ''n''-balls (criterion 3); or by letting f in criterion 4 be the multivariate normal density, for which \int f\,\text\mu=1.


Counting measure

Alternatively, consider the real numbers with the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
; the measure of any finite set is the number of elements in the set, and the measure of any infinite set is infinity. This measure 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. But, the set of natural numbers \mathbb N with the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
is ''σ'' -finite.


Locally compact groups

Locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
s which are σ-compact are σ-finite under the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...
. For example, all connected, locally compact groups ''G'' are σ-compact. To see this, let ''V'' be a relatively compact, symmetric (that is ''V'' = ''V''−1) open neighborhood of the identity. Then : H = \bigcup_ V^n is an open subgroup of ''G''. Therefore ''H'' is also closed since its complement is a union of open sets and by connectivity of ''G'', must be ''G'' itself. Thus all connected
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s are σ-finite under Haar measure.


Nonexamples

Any non-trivial measure taking only the two values 0 and \infty is clearly non σ-finite. One example in \R is: for all A \subset \R, \mu(A) = \infty if and only if A is not empty; another one is: for all A \subset \R, \mu(A) = \infty if and only if A is uncountable, 0 otherwise. Incidentally, both are translation-invariant.


Properties

The class of σ-finite measures has some very convenient properties; σ-finiteness can be compared in this respect to separability of topological spaces. Some theorems in analysis require σ-finiteness as a hypothesis. Usually, both the
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
and Fubini's theorem are stated under an assumption of σ-finiteness on the measures involved. However, as shown by Irving Segal, they require only a weaker condition, namely ''localisability''. Though measures which are not ''σ''-finite are sometimes regarded as pathological, they do in fact occur quite naturally. For instance, if ''X'' is a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
of
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
''r'', then all lower-dimensional
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assi ...
s are non-σ-finite if considered as measures on ''X''.


Equivalence to a probability measure

Any σ-finite measure ''μ'' on a space ''X'' is equivalent to a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
on ''X'': let ''V''''n'', ''n'' âˆˆ N, be a covering of ''X'' by pairwise disjoint measurable sets of finite ''μ''-measure, and let ''w''''n'', ''n'' âˆˆ N, be a sequence of positive numbers (weights) such that :\sum_^\infty w_n = 1. The measure ''ν'' defined by :\nu(A) = \sum_^\infty w_n \frac is then a probability measure on ''X'' with precisely the same
null set In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notio ...
s as ''μ''.


Related concepts


Moderate measures

A
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
(in the sense of a
locally finite measure In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Definition Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contain ...
on the Borel \sigma-algebra) \mu is called a moderate measure iff there are at most countably many open sets A_1, A_2, \ldots with \mu\left(A_i\right) < \infty for all i and \bigcup_^\infty A_i = X. Every moderate measure is a \sigma -finite measure, the converse is not true.


Decomposable measures

A measure is called a decomposable measure there are disjoint measurable sets \left(A_i\right)_ with \mu\left(A_i\right) < \infty for all i \in I and \bigcup_ A_i = X . For decomposable measures, there is no restriction on the number of measurable sets with finite measure. Every \sigma -finite measure is a decomposable measure, the converse is not true.


s-finite measures

A measure \mu is called a s-finite measure if it is the sum of at most countably many finite measures. Every σ-finite measure is s-finite, the converse is not true. For a proof and counterexample see relation to σ-finite measures.


See also

* Finite measure *
Sigma additivity In mathematics, an additive set function is a function \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this ad ...


References

{{DEFAULTSORT:Sigma finite measure Measures (measure theory)