Set function

In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \{ \pm \infty \}, which consists of the real numbers \R and \pm \infty.

A set function generally aims to measure 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 \mathcal{F} is a family of sets over \Omega (meaning that \mathcal{F} \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a set function on \mathcal{F} is a function \mu with domain \mathcal{F} and codomain [-\infty, \infty] 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 \mathcal{F}, or equivalently, that \mu does not take on both - \infty and + \infty 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) \text{ whenever } \mu(F) - \mu(E) is defined with E, F \in \mathcal{F} satisfying E \subseteq F and F \setminus E \in \mathcal{F}.

Null sets

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

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

Variation and mass

The Total variation S is

|\mu|(S) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \sup \{ |\mu(F)| : F \in \mathcal{F} \text{ and } F \subseteq S \}

where |\,\cdot\,| 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 \mathcal{F} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \textstyle\bigcup\limits_{F \in \mathcal{F}} F \in \mathcal{F}, then |\mu|\left(\cup \mathcal{F}\right) is called the of \mu and \mu\left(\cup \mathcal{F}\right) is called the of \mu.

A set function is called if for every F \in \mathcal{F}, the value \mu(F) is (which by definition means that \mu(F) \neq \infty and \mu(F) \neq -\infty; an is one that is equal to \infty or - \infty). Every finite set function must have a finite mass.

Common properties of set functions

A set function \mu on \mathcal{F} is said to be^[1]

if it is valued in [0, \infty].
Finitely additive set function if \textstyle\sum\limits_{i=1}^n \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^n F_i\right) for all pairwise disjoint finite sequences F_1, \ldots, F_n \in \mathcal{F} such that \textstyle\bigcup\limits_{i=1}^n F_i \in \mathcal{F}.
- If \mathcal{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 \mathcal{F}.
- If \mu is finitely additive and if \varnothing \in \mathcal{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 \infty, where in the latter case, \mu(E) = \mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing) = \mu(E) + (\pm \infty) = \pm \infty for every E \in \mathcal{F} (so only the case \mu(\varnothing) = 0 is useful).
Sigma-additive set function or Sigma-additive set function^[2] if in addition to being finitely additive, for all pairwise disjoint sequences F_1, F_2, \ldots\, in \mathcal{F} such that \textstyle\bigcup\limits_{i=1}^\infty F_i \in \mathcal{F}, all of the following hold:
1. \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)
  - The series on the left hand side is defined in the usual way as the limit \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\displaystyle\lim_{n \to \infty}} \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).
  - As a consequence, if \rho : \N \to \N is any permutation/bijection then \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \textstyle\sum\limits_{i=1}^\infty \mu\left(F_{\rho(i)}\right); this is because \textstyle\bigcup\limits_{i=1}^\infty F_i = \textstyle\bigcup\limits_{i=1}^\infty F_{\rho(i)} and applying this condition (a) twice guarantees that both \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) and \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_{\rho(i)}\right) = \textstyle\sum\limits_{i=1}^\infty \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\text{def}}{=}~ \textstyle\bigcup\limits_{i \in \N} 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) + \cdots and \mu(F) = \mu\left(F_{\rho(1)}\right) + \mu\left(F_{\rho(2)}\right) + \cdots\,.
2. if \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) is not infinite then this series \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) must also converge absolutely, which by definition means that \textstyle\sum\limits_{i=1}^\infty \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 \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = {\displaystyle\lim_{N \to \infty}} \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 [-\infty, \infty].
3. if \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) = \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) is infinite then it is also required that the value of at least one of the series \textstyle\sum\limits_{\stackrel{i \in \N}{\mu\left(F_i\right) > 0}} \mu\left(F_i\right) \; \text{ and } \; \textstyle\sum\limits_{\stackrel{i \in \N}{\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 Pre-measure if it is non-negative, countably additive (including finitely additive), and has a null empty set.
a Measure 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 Probability measure if it is a measure that has a mass of 1.
an Outer measure 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 Signed measure if it is countably additive, has a null empty set, and \mu does not take on both - \infty and + \infty as values.
Complete measure if every subset of every null set is null; explicitly, this means: whenever F \in \mathcal{F} \text{ satisfies } \mu(F) = 0 and N \subseteq F is any subset of F then N \in \mathcal{F} and \mu(N) = 0.
- Unlike many other properties, completeness places requirements on the set \operatorname{domain} \mu = \mathcal{F} (and not just on \mu's values).
σ-finite measure if there exists a sequence F_1, F_2, F_3, \ldots\, in \mathcal{F} such that \mu\left(F_i\right) is finite for every index i, and also \textstyle\bigcup\limits_{n=1}^\infty F_n = \textstyle\bigcup\limits_{F \in \mathcal{F}} F.
Decomposable measure if there exists a subfamily \mathcal{P} \subseteq \mathcal{F} of pairwise disjoint sets such that \mu(P) is finite for every P \in \mathcal{P} and also \textstyle\bigcup\limits_{P \in \mathcal{P}} \, P = \textstyle\bigcup\limits_{F \in \mathcal{F}} F (where \mathcal{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 Vector measure if it is a countably additive set function \mu : \mathcal{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, \|\cdot\|) then it is countably additive if and only if for any pairwise disjoint sequence F_1, F_2, \ldots\, in \mathcal{F}, \lim_{n \to \infty} \left\|\mu\left(F_1\right) + \cdots + \mu\left(F_n\right) - \mu\left(\textstyle\bigcup\limits_{i=1}^\infty 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 \mathcal{F}, \lim_{n \to \infty} \left\|\mu\left(F_n \cup F_{n+1} \cup F_{n+2} \cup \cdots\right)\right\| = 0.
a Complex measure if it is a countably additive complex-valued set function \mu : \mathcal{F} \to \Complex whose domain is a σ-algebra.
- By definition, a complex measure never takes \pm \infty as a value and so has a null empty set.
a Random measure 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 \in I} of real numbers indexed by an arbitrary indexing set I, it is possible to define their sum \textstyle\sum\limits_{i \in I} r_i as the limit of the net of finite partial sums F \in \operatorname{FiniteSubsets}(I) \mapsto \textstyle\sum\limits_{i \in F} r_i where the domain \operatorname{FiniteSubsets}(I) is directed by \,\subseteq.\, Whenever this net converges then its limit is denoted by the symbols \textstyle\sum\limits_{i \in I} r_i while if this net instead diverges to \pm \infty then this may be indicated by writing \textstyle\sum\limits_{i \in I} r_i = \pm \infty. Any sum over the empty set is defined to be zero; that is, if I = \varnothing then \textstyle\sum\limits_{i \in \varnothing} r_i = 0 by definition.

For example, if z_i = 0 for every i \in I then \textstyle\sum\limits_{i \in I} z_i = 0. And it can be shown that \textstyle\sum\limits_{i \in I} r_i = \textstyle\sum\limits_{\stackrel{i \in I,}{r_i = 0}} r_i + \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i = 0 + \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i = \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i. If I = \N then the generalized series \textstyle\sum\limits_{i \in I} r_i converges in \R if and only if \textstyle\sum\limits_{i=1}^\infty r_i converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series \textstyle\sum\limits_{i \in I} r_i converges in \R then both \textstyle\sum\limits_{\stackrel{i \in I}{r_i > 0}} r_i and \textstyle\sum\limits_{\stackrel{i \in I}{r_i < 0}} r_i also converge to elements of \R and the set \left\{i \in I : r_i \neq 0\right\} is necessarily countable (that is, either finite or countably infinite); this remains true if \R is replaced with any normed space.^{[proof 1]} It follows that in order for a generalized series \textstyle\sum\limits_{i \in 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 \textstyle\sum\limits_{i \in I} r_i ~=~ \textstyle\sum\limits_{\stackrel{i \in I}{r_i \neq 0}} r_i is a sum of at most countably many non-zero terms. Said differently, if \left\{i \in I : r_i \neq 0\right\} is uncountable then the generalized series \textstyle\sum\limits_{i \in 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 \mathcal{F} (and the usual countable series \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right)) to arbitrarily many sets \left(F_i\right)_{i \in I} (and the generalized series \textstyle\sum\limits_{i \in I} \mu\left(F_i\right)).

Inner measures, outer measures, and other properties

A set function \mu is said to be/satisfies^[1]

if \mu(E) \leq \mu(F) whenever E, F \in \mathcal{F} satisfy E \subseteq F.
Modular set function if it satisfies the following condition, known as modularity: \mu(E \cup F) + \mu(E \cap F) = \mu(E) + \mu(F) for all E, F \in \mathcal{F} such that E \cup F, E \cap F \in \mathcal{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 valuation. This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.
Submodular set function if \mu(E \cup F) + \mu(E \cap F) \leq \mu(E) + \mu(F) for all E, F \in \mathcal{F} such that E \cup F, E \cap F \in \mathcal{F}.
if |\mu(F)| \leq \textstyle\sum\limits_{i=1}^n \left|\mu\left(F_i\right)\right| for all finite sequences F, F_1, \ldots, F_n \in \mathcal{F} that satisfy F \;\subseteq\; \textstyle\bigcup\limits_{i=1}^n F_i.
or if |\mu(F)| \leq \textstyle\sum\limits_{i=1}^\infty \left|\mu\left(F_i\right)\right| for all sequences F, F_1, F_2, F_3, \ldots\, in \mathcal{F} that satisfy F \;\subseteq\; \textstyle\bigcup\limits_{i=1}^\infty F_i.
- If \mathcal{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 \mathcal{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(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) \leq \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) for all F_1, F_2, F_3, \ldots\, in \mathcal{F}.^[3] If \mu is a probability measure then this inequality is Boole's inequality.
- If \mu is countably subadditive and \varnothing \in \mathcal{F} with \mu(\varnothing) = 0 then \mu is finitely subadditive.
Superadditivity if \mu(E) + \mu(F) \leq \mu(E \cup F) whenever E, F \in \mathcal{F} are disjoint with E \cup F \in \mathcal{F}.
if \lim_{n \to \infty} \mu\left(F_i\right) = \mu\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right) for all non-increasing sequences of sets F_1 \supseteq F_2 \supseteq F_3 \cdots\, in \mathcal{F} such that \textstyle\bigcap\limits_{i=1}^\infty F_i \in \mathcal{F} with \mu\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right) and all \mu\left(F_i\right) finite.
- Lebesgue measure \lambda 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, \infty) so that \lim_{n \to \infty} \lambda\left(F_i\right) = \lim_{n \to \infty} \infty = \infty \neq 0 = \lambda(\varnothing) = \lambda\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right) where \textstyle\bigcap\limits_{i=1}^\infty F_i = \varnothing.
if \lim_{n \to \infty} \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) for all non-decreasing sequences of sets F_1 \subseteq F_2 \subseteq F_3 \cdots\, in \mathcal{F} such that \textstyle\bigcup\limits_{i=1}^\infty F_i \in \mathcal{F}.
if whenever F \in \mathcal{F} satisfies \mu(F) = \infty then for every real r > 0, there exists some F_r \in \mathcal{F} such that F_r \subseteq F and r \leq \mu\left(F_r\right) < \infty.
an outer measure if \mu is non-negative, countably subadditive, has a null empty set, and has the power set \wp(\Omega) as its domain.
an Inner measure if \mu is non-negative, superadditive, continuous from above, has a null empty set, has the power set \wp(\Omega) as its domain, and +\infty is approached from below.
atomic if every measurable set of positive measure contains an atom.

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

Translation invariant if \mu(\omega + F) = \mu(F) for all \omega \in \Omega and F \in \mathcal{F} such that \omega + F \in \mathcal{F}.

Topology related definitions

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

a Borel measure 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 Baire measure if it is a measure defined on the σ-algebra of all Baire sets.
Locally finite measure if for every point \omega \in \Omega there exists some neighborhood U \in \mathcal{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.
τ-additivity if \mu\left({\textstyle\bigcup} \, \mathcal{D}\right) = \sup_{D \in \mathcal{D}} \mu(D) whenever \mathcal{D} \subseteq \tau \cap \mathcal{F} is directed with respect to \,\subseteq\, and satisfies {\textstyle\bigcup} \, \mathcal{D} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \textstyle\bigcup\limits_{D \in \mathcal{D}} D \in \mathcal{F}.
- \mathcal{D} is directed with respect to \,\subseteq\, if and only if it is not empty and for all A, B \in \mathcal{D} there exists some C \in \mathcal{D} such that A \subseteq C and B \subseteq C.
Inner regular measure or if for every F \in \mathcal{F}, \mu(F) = \sup \{\mu(K) : F \supseteq K \text{ with } K \in \mathcal{F} \text{ a compact subset of } (\Omega, \tau)\}.
Outer regular measure if for every F \in \mathcal{F}, \mu(F) = \inf \{\mu(U) : F \subseteq U \text{ and } U \in \mathcal{F} \cap \tau\}.
Regular measure if it is both inner regular and outer regular.
a Borel regular measure if it is a Borel measure that is also regular.
a Radon measure if it is a regular and locally finite measure.
Strictly positive measure if every non-empty open subset has (strictly) positive measure.
a Valuation if it is non-negative, monotone, modular, has a null empty set, and has domain \tau.

Relationships between set functions

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

\mu is said to be with respect to \nu or dominated by \nu, 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 \frac{d \mu}{d \nu} exists and for every measurable F,
```
\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.
```

\mu and \nu are called Equivalence if each one is absolutely continuous with respect to the other. \mu is called a Equivalence of a measure \nu if \mu is \sigma-finite and they are equivalent.^[4]

\mu and \nu are Singular measure, 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) = \lim_{n \to \infty} \frac{|A \cap \{1, \ldots, n\}|}{n},

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 random set 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.^[5]

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 \lambda^{\!*\!} : \wp(\Reals) \to [0, \infty] that sends a subset E \subseteq \Reals to the infimum

\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \operatorname{length}(I_k) : {(I_k)_{k \in \N}} \text{ is a sequence of open intervals with } E \subseteq \bigcup_{k=1}^\infty I_k\right\}.

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:

\lambda^{\!*\!}(M) = \lambda^{\!*\!}(M \cap E) + \lambda^{\!*\!}(M \cap E^c) \quad \text{ for every } S \subseteq \Reals

is a measure that called Lebesgue measure. Vitali sets are examples of non-measurable sets of real numbers.

Infinite-dimensional space

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, \infty] that is identically equal to 0 (that is, it sends every S \subseteq \Reals to 0)^[6] 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.^[7]

Extending set functions

Extending from semialgebras to algebras

Suppose that \mu is a set function on a semialgebra \mathcal{F} over \Omega and let

\operatorname{algebra}(\mathcal{F}) := \left\{ F_1 \sqcup \cdots \sqcup F_n : n \in \N \text{ and } F_1, \ldots, F_n \in \mathcal{F} \text{ are pairwise disjoint } \right\},

which is the algebra on \Omega generated by \mathcal{F}. The archetypal example of a semialgebra that is not also an algebra is the family

\mathcal{S}_d := \{ \varnothing \} \cup \left\{ \left(a_1, b_1\right] \times \cdots \times \left(a_1, b_1\right] ~:~ -\infty \leq a_i < b_i \leq \infty \text{ for all } i = 1, \ldots, d \right\}

on \Omega := \R^d where (a, b] := \{ x \in \R : a < x \leq b \} for all -\infty \leq a < b \leq \infty.^[8] Importantly, the two non-strict inequalities \,\leq\, in -\infty \leq a_i < b_i \leq \infty cannot be replaced with strict inequalities \,<\, since semialgebras must contain the whole underlying set \R^d; that is, \R^d \in \mathcal{S}_d is a requirement of semialgebras (as is \varnothing \in \mathcal{S}_d).

If \mu is finitely additive then it has a unique extension to a set function \overline{\mu} on \operatorname{algebra}(\mathcal{F}) defined by sending F_1 \sqcup \cdots \sqcup F_n \in \operatorname{algebra}(\mathcal{F}) (where \,\sqcup\, indicates that these F_i \in \mathcal{F} are pairwise disjoint) to:^[8]

\overline{\mu}\left(F_1 \sqcup \cdots \sqcup F_n\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).

This extension \overline{\mu} will also be finitely additive: for any pairwise disjoint A_1, \ldots, A_n \in \operatorname{algebra}(\mathcal{F}), ^[8]

\overline{\mu}\left(A_1 \cup \cdots \cup A_n\right) = \overline{\mu}\left(A_1\right) + \cdots + \overline{\mu}\left(A_n\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}(\mathcal{F}) such that A \subseteq A_1 \cup \cdots \cup A_n,^[8]

\overline{\mu}\left(A\right) \leq \overline{\mu}\left(A_1\right) + \cdots + \overline{\mu}\left(A_n\right).

Extending from rings to σ-algebras

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

\mu^*(T) = \inf \left\{\sum_n \mu\left(S_n\right) : T \subseteq \cup_n S_n \text{ with } S_1, S_2, \ldots \in \mathcal{F}\right\}

and then restrict it to the set \mathcal{F}_M of \mu^*-measurable sets (that is, Carathéodory-measurable sets), which is the set of all M \subseteq \Omega such that

\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm{c}) \quad \text{ for every subset } S \subseteq \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, \infty] 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 \mu^*–measurable or Carathéodory-measurable if it satisfies the following Carathéodory's criterion:

\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm{c}) \quad \text{ for every subset } S \subseteq \Omega,

where M^\mathrm{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

^ Durrett 2019, pp. 1–37, 455–470.
^ Durrett 2019, pp. 466–470.
^ Royden & Fitzpatrick 2010, p. 30.
^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Vol. 77. Probability Theory and Stochastic Modelling. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
^ Kolmogorov and Fomin 1975
^ Rudin 1991, p. 139.
^ Rudin 1991, pp. 139–140.
^ Durrett 2019, pp. 1–9.

Proofs

^ ProofCountablyManyNon0Terms

References

A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0