Set function explained

\R \cup \{ \pm inf ty \}, which consists of the real numbers \R and \pm inf ty.

A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.

Definitions

If

l{F} is a family of sets over \Omega (meaning that l{F} \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain l{F} and codomain [ -inf ty, inf ty] or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.

In general, it is typically assumed that

\mu(E) + \mu(F) is always well-defined for all E, F \in l{F}, or equivalently, that \mu does not take on both - inf ty and + inf ty as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever \mu is finitely additive: \mu(F) - \mu(E) = \mu(F \setminus E) whenever \mu(F) - \mu(E) is defined with E, F \in l{F} satisfying E \subseteq F and F \setminus E \in l{F}.

Null sets

A set

F \in l{F} is called a (with respect to \mu ) or simply if \mu(F) = 0 . Whenever \mu is not identically equal to either -inf ty or +inf ty then it is typically also assumed that:

• : \mu(\varnothing) = 0 if \varnothing \in l{F}.

Variation and mass

S is$$|\backslash mu|(S)\; ~\backslash stackrel~\; \backslash sup\; \backslash$$where | ⋅ | denotes the absolute value (or more generally, it denotes the norm or seminorm if \mu is vector-valued in a (semi)normed space). Assuming that \cup l{F} ~\stackrel{\scriptscriptstyle def }~ \textstyle\bigcup\limits_ F \in \mathcal, then |\mu|\left(\cup l{F}\right) is called the of \mu and \mu\left(\cup l{F}\right) is called the of \mu.

A set function is called if for every

F \in l{F}, the value \mu(F) is (which by definition means that \mu(F) ≠ inf ty and \mu(F) ≠ -inf ty ; an is one that is equal to inf ty or - inf ty ). Every finite set function must have a finite mass.

Common properties of set functions

A set function

\mu on l{F} is said to be

• if it is valued in [ 0 , inf ty].
• if style\sum\limits ^{n} _{i =1} \mu\left(F _{i\right)} = \mu\left( style c up\limits ^{n} _{i =1} F _{i\right)} for all pairwise disjoint finite sequences F _{1,} \ldots, F _{n} \in l{F} such that style c up\limits ^{n} _{i =1} F _{i} \in l{F}.
  • If
  l{F} is closed under binary unions then \mu is finitely additive if and only if \mu(E \cup F) = \mu(E) + \mu(F) for all disjoint pairs E, F \in l{F}.
  • If
  \mu is finitely additive and if \varnothing \in l{F} then taking E : = F : = \varnothing shows that \mu(\varnothing) = \mu(\varnothing) + \mu(\varnothing) which is only possible if \mu(\varnothing) = 0 or \mu(\varnothing) = \pm inf ty, where in the latter case, \mu(E) = \mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing) = \mu(E) + (\pm inf ty) = \pm inf ty for every E \in l{F} (so only the case \mu(\varnothing) = 0 is useful).
• or if in addition to being finitely additive, for all pairwise disjoint sequences F _{1,} F _{2,} \ldots in l{F} such that style c up\limits ^{inf ty} _{i =1} F _{i} \in l{F}, all of the following hold:
  1. style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} = \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)}
     • The series on the left hand side is defined in the usual way as the limit
     style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} ~\stackrel{\scriptscriptstyle def }~ \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).
     • As a consequence, if
     \rho : \N \to \N is any permutation/bijection then style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} = style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{\rho(i)} \right); this is because style c up\limits ^{inf ty} _{i =1} F _{i} = style c up\limits ^{inf ty} _{i =1} F _{\rho(i)} and applying this condition (a) twice guarantees that both style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} = \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)} and \mu\left( style c up\limits ^{inf ty} _{i =1} F _{\rho(i)} \right) = style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{\rho(i)} \right) hold. By definition, a convergent series with this property is said to be unconditionally convergent. Stated in plain English, this means that rearranging/relabeling the sets F _{1,} F _{2,} \ldots to the new order F _{\rho(1)} , F _{\rho(2)} , \ldots does not affect the sum of their measures. This is desirable since just as the union F ~\stackrel{\scriptscriptstyle def }~ \textstyle\bigcup\limits_ F_i does not depend on the order of these sets, the same should be true of the sums \mu(F) = \mu\left(F _{1\right)} + \mu\left(F _{2\right)} + … and \mu(F) = \mu\left(F _{\rho(1)} \right) + \mu\left(F _{\rho(2)} \right) + … .
  2. if \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)} is not infinite then this series style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} must also converge absolutely, which by definition means that style\sum\limits ^{inf ty} _{i =1} \left|\mu\left(F _{i\right)\right|} must be finite. This is automatically true if \mu is non-negative (or even just valued in the extended real numbers).
     • As with any convergent series of real numbers, by the Riemann series theorem, the series
     style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} = {\displaystyle\lim _{N} } \mu\left(F_1\right) + \mu\left(F_2\right) + \cdots + \mu\left(F_N\right) converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if \mu is valued in [ -inf ty, inf ty].
  3. if \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)} = style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} is infinite then it is also required that the value of at least one of the series style\sum\limits _{\stackrel{i} {\mu\left(F _{i\right)} > 0 }} \mu\left(F _{i\right)} and style\sum\limits _{\stackrel{i} {\mu\left(F _{i\right)} < 0 }} \mu\left(F _{i\right)} be finite (so that the sum of their values is well-defined). This is automatically true if \mu is non-negative.
• a if it is non-negative, countably additive (including finitely additive), and has a null empty set.
• a if it is a pre-measure whose domain is a σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a null empty set.
• a if it is a measure that has a mass of 1 .
• an if it is non-negative, countably subadditive, has a null empty set, and has the power set \wp(\Omega) as its domain.
  • Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets

• a if it is countably additive, has a null empty set, and
  \mu does not take on both - inf ty and + inf ty as values.
• if every subset of every null set is null; explicitly, this means: whenever F \in l{F} satisfies \mu(F) = 0 and N \subseteq F is any subset of F then N \in l{F} and \mu(N) = 0 .
  • Unlike many other properties, completeness places requirements on the set
  \operatorname{domain} \mu = l{F} (and not just on \mu 's values).
• if there exists a sequence F _{1,} F _{2,} F _{3,} \ldots in l{F} such that \mu\left(F _{i\right)} is finite for every index i, and also style c up\limits ^{inf ty} _{n =1} F _{n} = style c up\limits _{F} } F.
• if there exists a subfamily l{P} \subseteq l{F} of pairwise disjoint sets such that \mu(P) is finite for every P \in l{P} and also style c up\limits _{P} } \, P = \textstyle\bigcup\limits_ F (where l{F} = \operatorname{domain} \mu ).
  • Every -finite set function is decomposable although not conversely. For example, the counting measure on
  \R (whose domain is \wp(\R) ) is decomposable but not -finite.
• a if it is a countably additive set function \mu : l{F} \to X valued in a topological vector space X (such as a normed space) whose domain is a σ-algebra.
  • If
  \mu is valued in a normed space (X, \| ⋅ \|) then it is countably additive if and only if for any pairwise disjoint sequence F _{1,} F _{2,} \ldots in l{F}, \lim _{n} \left\|\mu\left(F _{1\right)} + … + \mu\left(F _{n\right)} - \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)\right\|} = 0 . If \mu is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence F _{1,} F _{2,} \ldots in l{F}, \lim _{n} \left\|\mu\left(F _{n} \cup F _{n+1} \cup F _{n+2} \cup … \right)\right\| = 0 .
• a if it is a countably additive complex-valued set function \mu : l{F} \to \Complex whose domain is a σ-algebra.
  • By definition, a complex measure never takes
  \pm inf ty as a value and so has a null empty set.
• a if it is a measure-valued random element.

Arbitrary sums

As described in this article's section on generalized series, for any family

\left(r _{i\right)} _{i} of real numbers indexed by an arbitrary indexing set I, it is possible to define their sum style\sum\limits _{i} r _{i} as the limit of the net of finite partial sums F \in \operatorname{FiniteSubsets}(I) \mapsto style\sum\limits _{i} r _{i} where the domain \operatorname{FiniteSubsets}(I) is directed by \subseteq. Whenever this net converges then its limit is denoted by the symbols style\sum\limits _{i} r _{i} while if this net instead diverges to \pm inf ty then this may be indicated by writing style\sum\limits _{i} r _{i} = \pm inf ty. Any sum over the empty set is defined to be zero; that is, if I = \varnothing then style\sum\limits _{i} r _{i} = 0 by definition.

For example, if

z _{i} = 0 for every i \in I then style\sum\limits _{i} z _{i} = 0 . And it can be shown that style\sum\limits _{i} r _{i} = style\sum\limits _{\stackrel{i} {r _{i} = 0 }} r _{i} + style\sum\limits _{\stackrel{i} {r _{i} ≠ 0 }} r _{i} = 0 + style\sum\limits _{\stackrel{i} {r _{i} ≠ 0 }} r _{i} = style\sum\limits _{\stackrel{i} {r _{i} ≠ 0 }} r _{i.} If I = \N then the generalized series style\sum\limits _{i} r _{i} converges in \R if and only if style\sum\limits ^{inf ty} _{i =1} r _{i} converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series style\sum\limits _{i} r _{i} converges in \R then both style\sum\limits _{\stackrel{i} {r _{i} > 0 }} r _{i} and style\sum\limits _{\stackrel{i} {r _{i} < 0 }} r _{i} also converge to elements of \R and the set \left\{i \in I : r _{i} ≠ 0 \right\} is necessarily countable (that is, either finite or countably infinite); this remains true if \R is replaced with any normed space. It follows that in order for a generalized series style\sum\limits _{i} r _{i} to converge in \R or \Complex, it is necessary that all but at most countably many r _{i} will be equal to 0 , which means that style\sum\limits _{i} r _{i} ~ = ~ style\sum\limits _{\stackrel{i} {r _{i} ≠ 0 }} r _{i} is a sum of at most countably many non-zero terms. Said differently, if \left\{i \in I : r _{i} ≠ 0 \right\} is uncountable then the generalized series style\sum\limits _{i} r _{i} does not converge.

In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of "countably additive" is rarely extended from countably many sets

F _{1,} F _{2,} \ldots in l{F} (and the usual countable series style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} ) to arbitrarily many sets \left(F _{i\right)} _{i} (and the generalized series style\sum\limits _{i} \mu\left(F _{i\right)} ).

Inner measures, outer measures, and other properties

A set function

\mu is said to be/satisfies

• if \mu(E) \leq \mu(F) whenever E, F \in l{F} satisfy E \subseteq F.
• if it satisfies the following condition, known as : \mu(E \cup F) + \mu(E \cap F) = \mu(E) + \mu(F) for all E, F \in l{F} such that E \cup F, E \cap F \in l{F}.
  • Every finitely additive function on a field of sets is modular.
  • In geometry, a set function valued in some abelian semigroup that possess this property is known as a . This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.

• if
  \mu(E \cup F) + \mu(E \cap F) \leq \mu(E) + \mu(F) for all E, F \in l{F} such that E \cup F, E \cap F \in l{F}.
• if |\mu(F)| \leq style\sum\limits ^{n} _{i =1} \left|\mu\left(F _{i\right)\right|} for all finite sequences F, F _{1,} \ldots, F _{n} \in l{F} that satisfy F \subseteq style c up\limits ^{n} _{i =1} F _{i.}
• or if |\mu(F)| \leq style\sum\limits ^{inf ty} _{i =1} \left|\mu\left(F _{i\right)\right|} for all sequences F, F _{1,} F _{2,} F _{3,} \ldots in l{F} that satisfy F \subseteq style c up\limits ^{inf ty} _{i =1} F _{i.}
  • If
  l{F} is closed under finite unions then this condition holds if and only if |\mu(F \cup G)| \leq| \mu(F)| + |\mu(G)| for all F, G \in l{F}. If \mu is non-negative then the absolute values may be removed.
  • If
  \mu is a measure then this condition holds if and only if \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)} \leq style\sum\limits ^{inf ty} _{i =1} \mu\left(F _{i\right)} for all F _{1,} F _{2,} F _{3,} \ldots in l{F}. If \mu is a probability measure then this inequality is Boole's inequality.
  • If
  \mu is countably subadditive and \varnothing \in l{F} with \mu(\varnothing) = 0 then \mu is finitely subadditive.
• if \mu(E) + \mu(F) \leq \mu(E \cup F) whenever E, F \in l{F} are disjoint with E \cup F \in l{F}.
• if \lim _{n} \mu\left(F _{i\right)} = \mu\left( style c ap\limits ^{inf ty} _{i =1} F _{i\right)} for all of sets F _{1} \supseteq F _{2} \supseteq F _{3} … in l{F} such that style c ap\limits ^{inf ty} _{i =1} F _{i} \in l{F} with \mu\left( style c ap\limits ^{inf ty} _{i =1} F _{i\right)} and all \mu\left(F _{i\right)} finite.
  • Lebesgue measure
  λ is continuous from above but it would not be if the assumption that all \mu\left(F _{i\right)} are eventually finite was omitted from the definition, as this example shows: For every integer i, let F _{i} be the open interval (i, inf ty) so that \lim _{n} λ \left(F _{i\right)} = \lim _{n} inf ty = inf ty ≠ 0 = λ (\varnothing) = λ \left( style c ap\limits ^{inf ty} _{i =1} F _{i\right)} where style c ap\limits ^{inf ty} _{i =1} F _{i} = \varnothing.
• if \lim _{n} \mu\left(F _{i\right)} = \mu\left( style c up\limits ^{inf ty} _{i =1} F _{i\right)} for all of sets F _{1} \subseteq F _{2} \subseteq F _{3} … in l{F} such that style c up\limits ^{inf ty} _{i =1} F _{i} \in l{F}.
• if whenever F \in l{F} satisfies \mu(F) = inf ty then for every real r > 0 , there exists some F _{r} \in l{F} such that F _{r} \subseteq F and r \leq \mu\left(F _{r\right)} < inf ty.
• an if \mu is non-negative, countably subadditive, has a null empty set, and has the power set \wp(\Omega) as its domain.
• an if \mu is non-negative, superadditive, continuous from above, has a null empty set, has the power set \wp(\Omega) as its domain, and +inf ty is approached from below.
• if every measurable set of positive measure contains an atom.

+ is defined, then a set function \mu is said to be

• if \mu(\omega + F) = \mu(F) for all \omega \in \Omega and F \in l{F} such that \omega + F \in l{F}.

Topology related definitions

If

\tau is a topology on \Omega then a set function \mu is said to be:

• a if it is a measure defined on the σ-algebra of all Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing \tau ).
• a if it is a measure defined on the σ-algebra of all Baire sets.
• if for every point \omega \in \Omega there exists some neighborhood U \in l{F} \cap \tau of this point such that \mu(U) is finite.
  • If
  \mu is a finitely additive, monotone, and locally finite then \mu(K) is necessarily finite for every compact measurable subset K.
• if \mu\left({ style c up} l{D}\right) = \sup _{D} } \mu(D) whenever l{D} \subseteq \tau \cap l{F} is directed with respect to \subseteq and satisfies { style c up} l{D} ~\stackrel{\scriptscriptstyle def }~ \textstyle\bigcup\limits_ D \in \mathcal.l{D} is directed with respect to \subseteq if and only if it is not empty and for all A, B \in l{D} there exists some C \in l{D} such that A \subseteq C and B \subseteq C.
• or if for every F \in l{F}, \mu(F) = \sup \{\mu(K) : F \supseteq K with K \in l{F} acompactsubsetof (\Omega, \tau)\}.
• if for every F \in l{F}, \mu(F) = inf \{\mu(U) : F \subseteq U and U \in l{F} \cap \tau\}.
• if it is both inner regular and outer regular.
• a if it is a Borel measure that is also .
• a if it is a regular and locally finite measure.
• if every non-empty open subset has (strictly) positive measure.
• a if it is non-negative, monotone, modular, has a null empty set, and has domain \tau.

Relationships between set functions

See also: Radon–Nikodym theorem and Lebesgue's decomposition theorem.

If

\mu and \nu are two set functions over \Omega, then:

• \mu is said to be or, written \mu \ll \nu, if for every set F that belongs to the domain of both \mu and \nu, if \nu(F) = 0 then \mu(F) = 0 .
  • If
  \mu and \nu are \sigma -finite measures on the same measurable space and if \mu \ll \nu, then the Radon–Nikodym derivative (d \mu)/(d \nu) exists and for every measurable F, $$\backslash mu(F)\; =\; \backslash int\_F\; \backslash frac\; d\; \backslash nu.$$
\mu and \nu are called if each one is absolutely continuous with respect to the other. \mu is called a of a measure \nu if \mu is \sigma -finite and they are equivalent.^[1]
• \mu and \nu are , written \mu \perp \nu, if there exist disjoint sets M and N in the domains of \mu and \nu such that M \cup N = \Omega, \mu(F) = 0 for all F \subseteq M in the domain of \mu, and \nu(F) = 0 for all F \subseteq N in the domain of \nu.

Examples

Examples of set functions include:

• The function $$d(A)\; =\; \backslash lim\_\; \backslash frac$$

, assigning densities to sufficiently well-behaved subsets A \subseteq \{ 1 , 2 , 3 , \ldots\}, is a set function.

• A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is

1 , with other sets given probabilities between 0 and 1 .

• A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
• A is a set-valued random variable. See the article random compact set.

The Jordan measure on

\Reals ^{n} is a set function defined on the set of all Jordan measurable subsets of \Reals ^{n;} it sends a Jordan measurable set to its Jordan measure.

Lebesgue measure

The Lebesgue measure on

\Reals is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue \sigma -algebra.^[2]

Its definition begins with the set

\operatorname{Intervals}(\Reals) of all intervals of real numbers, which is a semialgebra on \Reals. The function that assigns to every interval I its \operatorname{length}(I) is a finitely additive set function (explicitly, if I has endpoints a \leq b then \operatorname{length}(I) = b - a ). This set function can be extended to the Lebesgue outer measure on \Reals, which is the translation-invariant set function λ ^{*} : \wp(\Reals) \to [ 0 , inf ty] that sends a subset E \subseteq \Reals to the infimum$$\backslash lambda^(E)\; =\; \backslash inf\; \backslash left\backslash .$$ Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the -algebra of all subsets M \subseteq \Reals that satisfy the Carathéodory criterion:$$\backslash lambda^(M)\; =\; \backslash lambda^(M\; \backslash cap\; E)\; +\; \backslash lambda^(M\; \backslash cap\; E^c)\; \backslash quad\; \backslash text\; S\; \backslash subseteq\; \backslash Reals$$is a measure that called Lebesgue measure. Vitali sets are examples of non-measurable sets of real numbers.

Infinite-dimensional space

See also: Abstract Wiener space, Feldman–Hájek theorem and Radonifying function.

As detailed in the article on infinite-dimensional Lebesgue measure, the only locally finite and translation-invariant Borel measure on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define Gaussian measures on infinite-dimensional topological vector spaces. The structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space.

Finitely additive translation-invariant set functions

The only translation-invariant measure on

\Omega = \Reals with domain \wp(\Reals) that is finite on every compact subset of \Reals is the trivial set function \wp(\Reals) \to [ 0 , inf ty] that is identically equal to 0 (that is, it sends every S \subseteq \Reals to 0 )However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in [ 0 , 1 ]. In fact, such non-trivial set functions will exist even if \Reals is replaced by any other abelian group G.

Extending set functions

See also: Carathéodory's extension theorem.

Extending from semialgebras to algebras

Suppose that

\mu is a set function on a semialgebra l{F} over \Omega and let $$\backslash operatorname(\backslash mathcal)\; :=\; \backslash left\backslash ,$$which is the algebra on \Omega generated by l{F}. The archetypal example of a semialgebra that is not also an algebra is the family$$\backslash mathcal\_d\; :=\; \backslash \; \backslash cup\; \backslash left\backslash$$on \Omega : = \R ^{d} where (a, b] : = \{ x \in \R : a < x \leq b \} for all -inf ty \leq a < b \leq inf ty. Importantly, the two non-strict inequalities \leq in -inf ty \leq a _{i} < b _{i} \leq inf ty cannot be replaced with strict inequalities < since semialgebras must contain the whole underlying set \R ^{d;} that is, \R ^{d} \in l{S} _{d} is a requirement of semialgebras (as is \varnothing \in l{S} _{d} ).

If

\mu is finitely additive then it has a unique extension to a set function \overline{\mu} on \operatorname{algebra}( l{F}) defined by sending F _{1} \sqcup … \sqcup F _{n} \in \operatorname{algebra}( l{F}) (where \sqcup indicates that these F _{i} \in l{F} are pairwise disjoint) to:$$\backslash overline\backslash left(F\_1\; \backslash sqcup\; \backslash cdots\; \backslash sqcup\; F\_n\backslash right)\; :=\; \backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash mu\backslash left(F\_n\backslash right).$$ This extension \overline{\mu} will also be finitely additive: for any pairwise disjoint A _{1,} \ldots, A _{n} \in \operatorname{algebra}( l{F}), $$\backslash overline\backslash left(A\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; A\_n\backslash right)\; =\; \backslash overline\backslash left(A\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash overline\backslash left(A\_n\backslash right).$$

If in addition

\mu is extended real-valued and monotone (which, in particular, will be the case if \mu is non-negative) then \overline{\mu} will be monotone and finitely subadditive: for any A, A _{1,} \ldots, A _{n} \in \operatorname{algebra}( l{F}) such that A \subseteq A _{1} \cup … \cup A _{n,} $$\backslash overline\backslash left(A\backslash right)\; \backslash leq\; \backslash overline\backslash left(A\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash overline\backslash left(A\_n\backslash right).$$

Extending from rings to σ-algebras

See also: Pre-measure.

If

\mu : l{F} \to [ 0 , inf ty] is a pre-measure on a ring of sets (such as an algebra of sets) l{F} over \Omega then \mu has an extension to a measure \overline{\mu} : \sigma( l{F}) \to [ 0 , inf ty] on the σ-algebra \sigma( l{F}) generated by l{F}. If \mu is σ-finite then this extension is unique.

To define this extension, first extend

\mu to an outer measure \mu ^{*} on 2 ^{\Omega} = \wp(\Omega) by $$\backslash mu^*(T)\; =\; \backslash inf\; \backslash left\backslash$$ and then restrict it to the set l{F} _{M} of \mu ^{*} -measurable sets (that is, Carathéodory-measurable sets), which is the set of all M \subseteq \Omega such that $$\backslash mu^*(S)\; =\; \backslash mu^*(S\; \backslash cap\; M)\; +\; \backslash mu^*(S\; \backslash cap\; M^\backslash mathrm)\; \backslash quad\; \backslash text\; S\; \backslash subseteq\; \backslash Omega.$$ It is a \sigma -algebra and \mu ^{*} is sigma-additive on it, by Caratheodory lemma.

Restricting outer measures

If

\mu ^{*} : \wp(\Omega) \to [ 0 , inf ty] is an outer measure on a set \Omega, where (by definition) the domain is necessarily the power set \wp(\Omega) of \Omega, then a subset M \subseteq \Omega is called or if it satisfies the following : $$\backslash mu^*(S)\; =\; \backslash mu^*(S\; \backslash cap\; M)\; +\; \backslash mu^*(S\; \backslash cap\; M^\backslash mathrm)\; \backslash quad\; \backslash text\; S\; \backslash subseteq\; \backslash Omega,$$where M ^{c} : = \Omega \setminus M is the complement of M.

The family of all

\mu ^{*} –measurable subsets is a σ-algebra and the restriction of the outer measure \mu ^{*} to this family is a measure.

Notes

Proofs

References

• • • A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover.

Further reading

• • Regular set function at Encyclopedia of Mathematics

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling . 77 . Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 21.
2. Kolmogorov and Fomin 1975