Signed measure

A signed measure is a generalization of the concept of a measure in measure theory, defined on a measurable space (X, \mathcal{A}) as a function \nu: \mathcal{A} \to \overline{\mathbb{R}} (the extended real line) such that \nu(\emptyset) = 0, \nu takes at most one of the values \pm \infty, and for any countable collection of pairwise disjoint sets \{E_n\}_{n=1}^\infty \subset \mathcal{A}, \nu\left(\bigcup_{n=1}^\infty E_n\right) = \sum_{n=1}^\infty \nu(E_n) (with the sum converging absolutely whenever it is finite).^[1]^[2] Signed measures extend positive measures by allowing negative values, enabling the representation of phenomena like net charges or differences in mass distributions, and they can be expressed as the difference of two positive measures under certain conditions, such as when at least one is finite.^[1]^[2] Key properties include continuity from below and above for monotone sequences of sets (with the latter requiring finite initial values), and the existence of positive sets (where subsets have non-negative measure) and negative sets (where subsets have non-positive measure).^[2] Central to the theory are the Hahn decomposition theorem, which partitions the space into a positive set and a negative set unique up to null sets, and the Jordan decomposition theorem, which uniquely expresses any signed measure \nu as \nu = \nu^+ - \nu^-, where \nu^+ and \nu^- are singular positive measures defining the total variation |\nu| = \nu^+ + \nu^-.^[1]^[2] These decompositions facilitate applications in integration theory, where signed measures arise naturally as \nu(E) = \int_E f \, d\mu for integrable functions f with respect to a positive measure \mu, and underpin the Radon-Nikodym theorem for densities between signed measures.^[1]

Definition and Basic Concepts

Formal Definition

A signed measure on a measurable space (X, \Sigma) is a function \nu: \Sigma \to \overline{\mathbb{R}} that is countably additive, satisfies \nu(\emptyset) = 0, and takes values in [-\infty, \infty) or (-\infty, \infty] but not both infinities simultaneously.^[1]^[3]^[2] The countable additivity axiom states that for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subset \Sigma,

\nu\left( \bigcup_{n=1}^\infty A_n \right) = \sum_{n=1}^\infty \nu(A_n),

where the series converges absolutely whenever the left-hand side is finite.^[1]^[3]^[2] Unlike positive measures, which map to [0, \infty] and model non-negative mass distributions, signed measures permit negative values, allowing representation of phenomena with opposing contributions such as electric charges or net mass flows.^[1]^[2] Such measures are often expressed notationally as \nu = \nu^+ - \nu^-, where the Hahn-Jordan decomposition theorem guarantees the existence of unique positive measures \nu^+ and \nu^- with disjoint supports satisfying this relation.^[3]^[2]

Hahn-Jordan Decomposition

The Hahn decomposition theorem provides a fundamental partition of the measurable space underlying a signed measure. Specifically, given a signed measure \nu on a measurable space (X, \Sigma), there exists a measurable set P \subseteq X such that \nu(A) \geq 0 for every measurable set A \subseteq P and \nu(B) \leq 0 for every measurable set B \subseteq X \setminus P. The set P is termed a positive set for \nu, while X \setminus P is a negative set. This decomposition splits X into regions where \nu behaves nonnegatively and nonpositively, respectively, and holds for any signed measure, including those that may take infinite values on certain sets provided the overall definition is satisfied. Building on the Hahn decomposition, the Jordan decomposition theorem establishes a canonical representation of the signed measure as a difference of two positive measures. Let P be a positive set from the Hahn decomposition; define the positive part \nu^+(E) = \nu(E \cap P) and the negative part \nu^-(E) = -\nu(E \cap (X \setminus P)) for any measurable E \subseteq X. Then \nu^+(E) \geq 0 and \nu^-(E) \geq 0, so \nu^+ and \nu^- are positive measures, \nu = \nu^+ - \nu^-, and \nu^+ and \nu^- are mutually singular (supported on disjoint sets P and X \setminus P, up to null sets). Moreover, this Jordan decomposition is unique: if \nu = \lambda - \mu for positive measures \lambda, \mu with \lambda \perp \mu, then \lambda = \nu^+ and \mu = \nu^-. The Hahn decomposition itself is unique up to sets of \nu-measure zero, meaning any two positive sets differ by a set N with \nu(N) = 0. From this, the total variation measure is given by |\nu|(E) = \nu^+(E) + \nu^-(E), which is a positive measure capturing the overall "size" of \nu. A key lemma for the Hahn decomposition states that if \nu(E) > 0 for some measurable E, then there exists a positive subset A \subseteq E with \nu(A) > 0. The proof of the Hahn decomposition proceeds as follows. Let \alpha = \sup \{ \nu(F) \mid F \in \Sigma, F \text{ is positive} \}, where \alpha \in [0, \infty]. There exists a sequence of positive sets \{A_n\}_{n=1}^\infty such that \nu(A_n) \to \alpha. Define P_n = \bigcup_{k=1}^n A_k, so \{P_n\} is increasing and positive, and set P = \bigcup_{n=1}^\infty P_n. By countable additivity and continuity from below, P is measurable and \nu(P) = \alpha. To show X \setminus P is negative, suppose there exists B \subseteq X \setminus P with \nu(B) > 0. By the lemma, there is a positive set A \subseteq B with \nu(A) > 0. Then P \cup A is positive (since A and P disjoint, and for any subset C \subseteq P \cup A, \nu(C) = \nu(C \cap P) + \nu(C \cap A) \geq 0) and \nu(P \cup A) = \nu(P) + \nu(A) > \alpha, contradicting the definition of \alpha. Thus, no such B exists.^[2]^[1]

Examples

Simple Examples

A basic example of a signed measure arises on a finite discrete space, such as the set X = \{1, 2, 3\} equipped with the power set \sigma-algebra. Define \nu: \mathcal{P}(X) \to \mathbb{R} by \nu(\{1\}) = 2, \nu(\{2\}) = -1, and \nu(\{3\}) = 0, extending additively to all subsets; for instance, \nu(\{1,2\}) = 1 and \nu(X) = 1.^[4] This \nu is a signed measure because it satisfies \nu(\emptyset) = 0 and countable additivity (which reduces to finite additivity here, as X is finite). In this case, the Hahn-Jordan decomposition partitions X into the positive set P = \{1\} and negative set N = \{2\}, with \nu^+(A) = \nu(A \cap P) and \nu^-(A) = -\nu(A \cap N).^[4] Another simple construction involves signed point masses, or Dirac signed measures, on \mathbb{R} with the Borel \sigma-algebra. Consider \nu = \delta_0 - \delta_1, where \delta_x(A) = 1 if x \in A and 0 otherwise for any Borel set A \subseteq \mathbb{R}. Then \nu(A) = 1 if $0 \in A and $1 \notin A, \nu(A) = -1 if $1 \in A and $0 \notin A, \nu(A) = 0 if both or neither are in A.^[5] This \nu is a signed measure, as it inherits countable additivity from the Dirac measures: for disjoint Borel sets A_n, \nu(\bigcup A_n) = \sum \nu(A_n), since at most one A_n can contain 0 or 1.^[5] Signed measures can also be defined using Lebesgue measure on intervals. On the Borel subsets of [0,2], let \nu(A) = \lambda(A \cap [0,1]) - \lambda(A \cap [1,2]), where \lambda is the Lebesgue measure. For example, \nu([0,0.5]) = 0.5, \nu([1.5,2]) = -0.5, and \nu([0,2]) = 0.^[4] Countable additivity holds because Lebesgue measure is \sigma-additive: if \{A_n\} are disjoint Borel subsets, then \nu(\bigcup A_n) = \lambda((\bigcup A_n) \cap [0,1]) - \lambda((\bigcup A_n) \cap [1,2]) = \sum \lambda(A_n \cap [0,1]) - \sum \lambda(A_n \cap [1,2]) = \sum \nu(A_n).^[4]

Constructions from Functions

A fundamental method to construct signed measures involves integrating a signed measurable function with respect to a positive measure. Specifically, given a measurable space (X, \mathcal{S}), a positive measure \mu on \mathcal{S}, and a signed measurable function f: X \to \mathbb{R} that is \mu-integrable (meaning \int_X |f| \, d\mu < \infty), the set function defined by \nu(A) = \int_A f \, d\mu for all A \in \mathcal{S} is a signed measure on \mathcal{S}.^[4]^[5] This construction leverages the countable additivity of \mu and the linearity of the integral, ensuring that \nu inherits the required properties while allowing negative values where f is negative.^[4] Every signed measure \nu that is absolutely continuous with respect to a positive measure \mu (i.e., \nu(A) = 0 whenever \mu(A) = 0) admits such a representation via the Radon–Nikodym theorem, provided \mu is \sigma-finite. In this case, there exists a unique (up to \mu-almost everywhere equality) \mu-integrable function f such that \nu(A) = \int_A f \, d\mu for all A \in \mathcal{S}.^[4] Furthermore, the Jordan decomposition of \nu aligns with the positive and negative parts of f: letting f^+ = \max(f, 0) and f^- = \max(-f, 0), the positive and negative parts are \nu^+(A) = \int_A f^+ \, d\mu and \nu^-(A) = \int_A f^- \, d\mu, respectively.^[5] This indefinite integral form provides a canonical way to represent and decompose signed measures in L^1(\mu).^[4] A concrete example illustrates this construction on the interval [0,2] with Lebesgue measure \lambda. Consider f(x) = x - 1 and define \nu(A) = \int_A (x - 1) \, d\lambda(x) for Borel sets A \subseteq [0,2]. Here, \nu is a signed measure absolutely continuous with respect to \lambda, positive on sets concentrated to the right of 1 (where f > 0) and negative on sets to the left (where f < 0), with total variation |\nu|(A) = \int_A |x - 1| \, d\lambda(x).^[5] The key condition for \nu to be a signed measure is the \mu-integrability of f, which guarantees finite values and countable additivity: for disjoint sets A_n, \nu(\bigcup A_n) = \sum \nu(A_n) follows from the corresponding property of the Lebesgue integral with respect to \mu. Without integrability, the set function may fail to be well-defined or additive.^[4]^[5]

Key Properties

Additivity and Continuity

A signed measure \nu on a measurable space (X, \Sigma) satisfies finite additivity: for any finite collection of pairwise disjoint sets A_1, \dots, A_n \in \Sigma, \nu\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n \nu(A_i).^[6] This property follows directly from the countable additivity axiom, as the finite union can be viewed as a countable union by including empty sets.^[7] The defining feature of a signed measure is countable additivity, also known as \sigma-additivity: for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subset \Sigma, \nu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \nu(A_n), and the series converges absolutely whenever the sum is finite.^[2] This extends the finite case and ensures the measure behaves consistently under countable disjoint unions, distinguishing signed measures from merely finitely additive set functions.^[8] Signed measures also exhibit continuity properties with respect to monotone limits of sets. Specifically, continuity from below holds: if A_n \uparrow A (i.e., A_n \subset A_{n+1} for all n and \bigcup_{n=1}^\infty A_n = A), then \nu(A_n) \uparrow \nu(A).^[7] Continuity from above holds under a finiteness condition: if A_n \downarrow A (i.e., A_{n+1} \subset A_n for all n and \bigcap_{n=1}^\infty A_n = A) and \nu(A_1) < \infty, then \nu(A_n) \downarrow \nu(A).^[6] These properties mirror those of positive measures and follow from countable additivity applied to the disjoint differences in the monotone sequences.^[8] Via the Hahn-Jordan decomposition, any signed measure \nu can be uniquely expressed as \nu = \nu^+ - \nu^-, where \nu^+ and \nu^- are nonnegative measures that are mutually singular.^[7] The positive and negative parts \nu^+ and \nu^- each inherit the additivity and continuity properties of \nu, and as nonnegative measures, they satisfy monotonicity: if A \subset B, then \nu^+(A) \leq \nu^+(B) and \nu^-(A) \leq \nu^-(B).^[6] This decomposition preserves the additive and limit behaviors while ensuring the parts are monotone increasing functions on nested sets.^[8]

Absolute Continuity and Singularity

In measure theory, a signed measure \nu on a measurable space (X, \mathcal{M}) is said to be absolutely continuous with respect to a positive measure \mu on the same space, denoted \nu \ll \mu, if for every measurable set A \in \mathcal{M} with \mu(A) = 0, it follows that \nu(A) = 0.^[9] This condition ensures that \nu does not charge any set of \mu-measure zero, capturing a form of dependence where the "size" of sets under \nu is controlled by \mu. For finite signed measures, this is equivalent to the \epsilon-\delta condition: for every \epsilon > 0, there exists \delta > 0 such that |\nu(A)| < \epsilon whenever \mu(A) < \delta.^[9] Given the Hahn-Jordan decomposition of \nu into its positive and negative parts \nu = \nu^+ - \nu^-, where \nu^+ and \nu^- are mutually singular positive measures, absolute continuity \nu \ll \mu holds if and only if both \nu^+ \ll \mu and \nu^- \ll \mu.^[10] This equivalence follows from the construction of the Jordan parts via a Hahn decomposition, where sets of \mu-measure zero cannot contribute to either part without violating the absolute continuity of \nu. Consequently, the total variation measure |\nu| = \nu^+ + \nu^- also satisfies |\nu| \ll \mu. Two measures \mu and \nu (where \nu may be signed) on (X, \mathcal{M}) are mutually singular, denoted \nu \perp \mu, if there exist disjoint measurable sets P, N \in \mathcal{M} such that P \cup N = X, \mu(P) = 0, and \nu(N) = 0.^[9] In this case, \nu is supported entirely on P (up to sets of \mu-measure zero), while \mu is supported on N, reflecting a complete lack of overlap in their "supports." For signed \nu, this extends the notion from positive measures by requiring that the entire signed measure vanishes on N, which aligns with the singularity of both \nu^+ and \nu^- with respect to \mu. The Lebesgue decomposition theorem provides a canonical way to break down any signed measure relative to a positive measure. Specifically, if \nu is a \sigma-finite signed measure and \mu is a \sigma-finite positive measure on (X, \mathcal{M}), then there exist unique \sigma-finite signed measures \nu_{ac} and \nu_s such that \nu = \nu_{ac} + \nu_s, with \nu_{ac} \ll \mu and \nu_s \perp \mu.^[9] This decomposition is unique up to sets of \mu-measure zero and generalizes the classical Lebesgue decomposition for positive measures by applying the Hahn-Jordan decomposition to the absolutely continuous and singular components. The absolutely continuous part \nu_{ac} captures the dependence on \mu, while \nu_s represents the independent, singular behavior. Under the absolute continuity condition \nu \ll \mu with \mu \sigma-finite, the Radon-Nikodym theorem extends to signed measures: there exists a \mu-integrable function f \in L^1(\mu) (taking both positive and negative values) such that for every measurable set A \in \mathcal{M},

\nu(A) = \int_A f \, d\mu.

^[9] This f = d\nu / d\mu, the Radon-Nikodym derivative, is unique up to \mu-almost everywhere equivalence and can be obtained by applying the theorem separately to \nu^+ and \nu^-, yielding f = f^+ - f^- where f^\pm \geq 0. Such representations connect signed measures to integrable functions with respect to \mu, as in the constructions from density functions.^[10]

Advanced Structures

Total Variation Measure

The total variation measure of a signed measure \nu on a measurable space (X, \Sigma) is defined for each A \in \Sigma by

|\nu|(A) = \sup\left\{ \sum_{i=1}^n |\nu(A_i)| : n \in \mathbb{N}, \{A_i\}_{i=1}^n \text{ is a finite partition of } A \right\},

where the supremum is taken over all finite partitions of A into measurable sets.^[1] This definition yields a positive measure |\nu| on (X, \Sigma), as it satisfies the axioms of a measure, including countable additivity.^[11] From the Hahn-Jordan decomposition \nu = \nu^+ - \nu^-, where \nu^+ and \nu^- are mutually singular positive measures, it follows that |\nu|(A) = \nu^+(A) + \nu^-(A) for all A \in \Sigma.^[12] The total variation |\nu| possesses several key properties that highlight its role in measure theory. It is the minimal positive measure \mu such that |\nu(A)| \leq \mu(A) for every A \in \Sigma, meaning no smaller positive measure dominates the absolute values of \nu.^[1] Additionally, \nu is absolutely continuous with respect to |\nu|, denoted \nu \ll |\nu|, which ensures that the Radon-Nikodym derivative of \nu with respect to |\nu| exists and equals the sign function of \nu almost everywhere with respect to |\nu|.^[11] These properties make |\nu| a canonical extension of \nu to a positive measure that captures its oscillatory behavior. The total variation norm of \nu is given by \|\nu\| = |\nu|(X), which is finite if and only if \nu is of bounded variation, i.e., |\nu|(X) < \infty.^[12] This norm provides a natural way to quantify the size of \nu and induces a Banach space structure on the set of signed measures of bounded variation. In applications, the total variation controls the differences in \nu: for any A, B \in \Sigma, |\nu(A) - \nu(B)| \leq |\nu|(A \Delta B), where A \Delta B is the symmetric difference, bounding how much \nu can change over sets.^[1]

The Space of Signed Measures

The space of finite signed measures on a measurable space (X, \Sigma), denoted M(X) or ca(X), forms a vector space with pointwise addition and scalar multiplication defined by (\nu + \lambda)(A) = \nu(A) + \lambda(A) and (c\nu)(A) = c \cdot \nu(A) for finite signed measures \nu, \lambda and scalar c \in \mathbb{R}, with all A \in \Sigma.^[13] This space, consisting of those satisfying \|\nu\| = |\nu|(X) < \infty where |\nu| denotes the total variation measure, is a Banach space under the total variation norm \|\nu\| = |\nu|(X).^[12] This norm, generated by the total variation measure, ensures completeness of the space.^[14] On locally compact Hausdorff spaces X, the space M(X) of finite signed Radon measures inherits this Banach space structure.^[15] The space M(X) admits a natural duality with the Banach space C_b(X) of bounded continuous real-valued functions on X, equipped with the supremum norm, via the pairing \langle f, \nu \rangle = \int_X f \, d\nu.^[16] The weak^* topology on M(X) is defined by this duality: a sequence \{\nu_n\} converges to \nu in the weak^* topology if and only if \int_X f \, d\nu_n \to \int_X f \, d\nu for every f \in C_b(X).^[17]