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Hahn decomposition theorem

The Hahn decomposition theorem is a foundational result in measure theory that provides a partition of a measurable space into positive and negative sets with respect to any given signed measure. A signed measure on a measurable space (X, \mathcal{M}) is a countably additive function \nu: \mathcal{M} \to [-\infty, \infty] with \nu(\emptyset) = 0 and taking at most one infinite value, extending the concept of positive measures to allow negative values. Named after the Austrian mathematician Hans Hahn, the theorem states that for any such signed measure \nu, there exist measurable sets P, N \in \mathcal{M} such that P \cup N = X, P \cap N = \emptyset, P is a positive set (i.e., \nu(E) \geq 0 for all measurable E \subseteq P), and N is a negative set (i.e., \nu(E) \leq 0 for all measurable E \subseteq N). This decomposition is unique up to sets of \nu-measure zero: if P' and N' form another such partition, then the symmetric differences P \Delta P' and N \Delta N' are null sets for \nu. The Hahn decomposition enables the Jordan decomposition of the signed measure \nu into positive and negative parts, defined as \nu^+(E) = \nu(E \cap P) and \nu^-(E) = -\nu(E \cap N), yielding \nu = \nu^+ - \nu^- where \nu^+ and \nu^- are mutually singular positive measures. These singular measures satisfy \nu^+(N) = 0 and \nu^-(P) = 0, ensuring the decomposition captures the "positive" and "negative" behaviors of \nu without overlap. The theorem's significance lies in its role as a cornerstone for advanced measure-theoretic tools, including the integration of signed measures via \int f \, d\nu = \int f \, d\nu^+ - \int f \, d\nu^- for suitable functions f, and as a prerequisite for the Radon–Nikodym theorem, which addresses densities between measures. It applies broadly in probability, functional analysis, and stochastic processes, where signed measures model phenomena like signed probabilities or differences of mass distributions. Proofs typically rely on Zorn's lemma to construct maximal positive sets from the partially ordered family of positive sets, ensuring the complement is negative.

Background Concepts

Signed Measures

A on a (X, \mathcal{M}), where \mathcal{M} is a \sigma-algebra of subsets of X, is a function \mu: \mathcal{M} \to \overline{\mathbb{R}} (the extended real numbers) that satisfies countable additivity: for any countable collection of pairwise disjoint sets \{E_n\}_{n=1}^\infty \subset \mathcal{M} whose union is also in \mathcal{M}, \mu\left( \bigcup_{n=1}^\infty E_n \right) = \sum_{n=1}^\infty \mu(E_n), where the series converges in the extended reals (possibly to \pm \infty), and \mu does not attain both +\infty and -\infty. Additionally, \mu(\emptyset) = 0, which follows directly from additivity by taking the empty collection. This definition extends the notion of a positive measure, which maps to [0, \infty] and is non-negative, by permitting negative values while preserving \sigma-additivity on disjoint sets; however, the restriction against both infinite signs ensures well-defined behavior under differences and sums. Signed measures commonly arise as differences of positive measures. For instance, if \nu and \lambda are positive measures on (X, \mathcal{M}), then \mu = \nu - \lambda defines a signed measure, provided \mu avoids taking both +\infty and -\infty (e.g., if one dominates the other on sets of positive measure). Another example is a signed : for a point x \in X and real coefficient c \in \mathbb{R}, define \mu(E) = c if x \in E and \mu(E) = 0 otherwise, which is \sigma-additive since disjoint sets containing x (at most one) yield the appropriate sum. A simple finite case occurs on the power set of a \{1, \dots, n\} with \mu(E) = \sum_{i \in E} a_i for real numbers a_i, which extends countably additive trivially as there are no infinite disjoint unions. Basic properties of signed measures include finite additivity as a consequence of \sigma-additivity, and under \sigma-finiteness—meaning X admits an increasing sequence of sets X_k \uparrow X with |\mu|(X_k) < \infty for a suitable total variation |\mu|—finite additivity implies full \sigma-additivity. A signed measure \mu is called positive if \mu(E) \geq 0 for all E \in \mathcal{M}, in which case it coincides with a positive measure, and negative if \mu(E) \leq 0 for all E \in \mathcal{M}. These properties ensure signed measures form a vector space over \mathbb{R}, closed under scalar multiplication by reals and addition of compatible signed measures.

Positive and Negative Sets

In the context of a signed measure \mu on a measurable space (X, \Sigma), sets are classified based on the sign of \mu restricted to their measurable subsets. A measurable set P \in \Sigma is called a positive set with respect to \mu if \mu(E) \geq 0 for every measurable set E \in \Sigma such that E \subseteq P. Similarly, a measurable set N \in \Sigma is a negative set if \mu(E) \leq 0 for every E \in \Sigma with E \subseteq N. A measurable set Z \in \Sigma is a null set if \mu(E) = 0 for every E \in \Sigma with E \subseteq Z; such sets are both positive and negative. These classifications exhibit hereditary and closure properties under basic set operations. Any measurable subset of a positive set is itself positive, as the restriction of \mu to subsets preserves the non-negativity condition. Moreover, the union of two positive sets is positive, since any measurable subset of the union can be decomposed into parts lying within each original positive set, where \mu remains non-negative. Analogous properties hold for negative sets: any measurable subset of a negative set is negative, and the union of two negative sets is negative. The empty set is both positive and negative for any signed measure. To illustrate these concepts, consider the interval [0,1] equipped with the Borel \sigma-algebra and Lebesgue measure \lambda, and let A \subseteq [0,1] be a fixed measurable set. Define the signed measure \mu(E) = \lambda(E) - 2\lambda(E \cap A) for Borel sets E \subseteq [0,1]. The complement [0,1] \setminus A is a positive set, as any Borel E \subseteq [0,1] \setminus A satisfies \mu(E) = \lambda(E) - 2\lambda(\emptyset) = \lambda(E) \geq 0. In contrast, A is a negative set, since any Borel E \subseteq A yields \mu(E) = \lambda(E) - 2\lambda(E) = -\lambda(E) \leq 0.

Statement of the Theorem

Formal Statement

The Hahn decomposition theorem states that for a measurable space (X, \Sigma) and a signed measure \mu: \Sigma \to [-\infty, \infty] on \Sigma (where \mu is \sigma-additive and does not take both +\infty and -\infty), there exist disjoint sets P, N \in \Sigma such that P \cup N = X, \mu(E) \geq 0 for every measurable set E \subseteq P, and \mu(E) \leq 0 for every measurable set E \subseteq N. Here, P is called a positive set for \mu and N is called a negative set for \mu. The decomposition X = P \sqcup N is unique up to \mu-null sets (i.e., if \{P', N'\} is another such pair, then P \triangle P' is \mu-null) provided that \mu is \sigma-finite. The positive and negative parts of \mu are then defined by \mu^+(E) = \mu(E \cap P), \quad \mu^-(E) = -\mu(E \cap N) for all E \in \Sigma, yielding the Jordan decomposition \mu = \mu^+ - \mu^-.

Interpretation and Intuition

The Hahn decomposition theorem addresses the inherent ambiguity in signed measures by partitioning the measurable space into two disjoint sets: a positive set, where the measure assigns non-negative values to all measurable subsets, and a negative set, where it assigns non-positive values. This "sign-separation" resolves the mixed behavior of signed measures, enabling their representation as the difference of two positive measures supported on these respective sets, thereby facilitating the application of techniques developed for positive measures alone. Signed measures emerge naturally in fields like potential theory, where they model the distribution of electric charges that can be positive or negative, as in the Poisson integral representation of harmonic functions derived from in electrostatics. In probability theory, they appear in contexts involving signed probabilities, such as modeling discrepancies or adjustments in expectation calculations. The theorem's decomposition simplifies analytical tasks by transforming signed measure problems into those involving positive measures, which possess desirable properties like monotonicity and subadditivity. A concrete illustration occurs on the interval [-1, 1] equipped with the Borel \sigma-algebra and restricted Lebesgue measure, where the signed measure \mu is defined by \mu(E) = \int_E x \, dx for Borel sets E \subseteq [-1, 1]. In this case, the positive set is [0, 1], as integrals over its subsets are non-negative due to the non-negative integrand, while the negative set is [-1, 0), where integrals are non-positive; these sets form a Hahn decomposition, yielding the Jordan decomposition \mu = \mu^+ - \mu^- with \mu^+ and \mu^- supported accordingly. The decomposition's uniqueness up to null sets—sets of \mu-measure zero—ensures that any two Hahn decompositions differ only on negligible portions of the space, preserving the essential structure. This property relies on the signed measure being \sigma-finite, meaning the space can be covered by countably many sets of finite measure, which prevents pathological behaviors in infinite spaces. Conceptually, the theorem parallels the decomposition of a linear functional on a vector space of functions into positive and negative components, mirroring how signed measures act as bounded linear functionals on spaces like L^\infty, with the positive and negative parts capturing the functional's directional behaviors.

Proof of the Hahn Decomposition Theorem

Construction of the Decomposition

The construction of the Hahn decomposition begins by applying to a partially ordered family of pairs of measurable sets. Consider the collection \mathcal{F} of all pairs (P, N) where P and N are measurable subsets of X such that P \cap N = \emptyset, P is positive for \nu (meaning \nu(E) \geq 0 for every measurable E \subseteq P), and N is negative for \nu (meaning \nu(E) \leq 0 for every measurable E \subseteq N). This collection is partially ordered by inclusion: (P_1, N_1) \preceq (P_2, N_2) if P_1 \subseteq P_2 and N_1 \subseteq N_2. The pair (\emptyset, \emptyset) belongs to \mathcal{F}, so the collection is nonempty. Any chain in \mathcal{F} has an upper bound given by the pointwise union of the P's and the pointwise union of the N's in the chain; the union of positive sets is positive, and the union of negative sets is negative, by countable additivity of the signed measure \nu. By , \mathcal{F} has a maximal element (P_0, N_0). To establish that P_0 \cup N_0 = X, suppose for contradiction that the set S = X \setminus (P_0 \cup N_0) is nonempty. Then there exists a measurable subset A \subseteq S with $0 < \nu(A) < \infty (if no such finite-measure subset exists, restrict iteratively to parts where \nu takes finite values). Maximality of (P_0, N_0) implies that no nontrivial positive or negative set can be extracted from A to extend the pair. To see this, define the oscillation of \nu over a measurable set B by \osc(\nu, B) = \sup \{ \nu(E) - \nu(F) : E, F \subseteq B,\, E \cap F = \emptyset,\, E, F \text{ measurable} \}. This quantity captures the extent to which \nu exhibits both positive and negative behavior on subsets of B; if \osc(\nu, B) = 0, then \nu(E) = \nu(F) for all disjoint measurable E, F \subseteq B, which implies \nu(G) = 0 for all measurable G \subseteq B (taking F = \emptyset or E = \emptyset). For the remaining set S, maximality ensures \osc(\nu, S) = 0, as a positive value would allow splitting S into disjoint E and F with \nu(E) > \nu(F). By the Hahn extension lemma, this yields a positive set Q \subseteq S with \nu(Q) > 0 and a negative set R \subseteq S with \nu(R) < 0, disjoint from Q, allowing an extension of the pair (e.g., (P_0 \cup Q, N_0 \cup R)), contradicting maximality. But \osc(\nu, A) \geq \nu(A) > 0 (taking E = A, F = \emptyset), yielding a contradiction unless no such A exists. Thus, S = \emptyset. If \nu is not \sigma-finite, the construction proceeds iteratively on a countable collection of disjoint measurable sets \{X_n\}_{n \in \mathbb{N}} such that \osc(\nu, X_n) < \infty for each n and X = \bigcup_n X_n \cup Z where Z has \osc(\nu, Z) = 0 (or is handled separately as a null set). Apply the above procedure to the restriction of \nu to each X_n, yielding decompositions (P_n, N_n) with P_n \cup N_n = X_n. The desired sets are then P_0 = \bigcup_n P_n and N_0 = \bigcup_n N_n \cup Z, which satisfy the properties by additivity of \nu.

Verification of Properties

To verify that the constructed sets P and N = X \setminus P form a , where P is a maximal positive set obtained via applied to the partially ordered collection of positive sets under inclusion, the following properties must hold for the signed measure \nu on the measurable space (X, \mathcal{M}). First, P is a positive set, meaning \nu(E) \geq 0 for every measurable E \subseteq P. By construction, P belongs to the collection of positive sets, so this holds directly; however, to confirm via maximality, suppose there exists a measurable E \subseteq P with \nu(E) < 0. Then, the implies there exists a negative measurable subset F \subseteq E with \nu(F) < 0. But P \setminus F would then be a positive set properly containing a subset of P while maintaining non-negativity on subsets, contradicting the maximality of P. Thus, no such E exists, verifying positivity. Symmetrically, N is a negative set, meaning \nu(E) \leq 0 for every measurable E \subseteq N. Suppose, for contradiction, there exists a measurable E \subseteq N with \nu(E) > 0. The Hahn lemma guarantees a positive measurable Q \subseteq E with \nu(Q) > 0. Then P \cup Q would be positive (as the union of two positive sets with disjoint supports relative to \nu) and strictly larger than P, contradicting maximality. Hence, no such E exists. The sets cover the space, as P \cup N = P \cup (X \setminus P) = X by definition. They are disjoint, since P \cap N = P \cap (X \setminus P) = \emptyset. Finally, the decomposition is unique up to \nu-null sets. Suppose (P', N') is another Hahn decomposition with N' = X \setminus P'. Consider P \setminus P', which is measurable and subsets of the positive set P, so \nu(F) \geq 0 for any measurable F \subseteq P \setminus P'. But P \setminus P' \subseteq N', so \nu(F) \leq 0 for all such F. Thus, \nu(F) = 0 for all F \subseteq P \setminus P', implying \nu(P \setminus P') = 0. Similarly, \nu(P' \setminus P) = 0, so the P \Delta P' is \nu-null, and the same holds for N \Delta N'. For null sets in the intersections, if E \subseteq P \cap N', then E is both positive and negative, forcing \nu(E) = 0.

Applications and Extensions

Jordan Measure Decomposition

The Hahn decomposition theorem provides a partition of the measurable space into a positive set P and a negative set N, which directly induces the Jordan decomposition of a \mu. Given such sets P and N with P \cup N = X and P \cap N = \emptyset, the positive part \mu^+ and negative part \mu^- are defined for any measurable set E by \mu^+(E) = \mu(E \cap P), \quad \mu^-(E) = -\mu(E \cap N). Both \mu^+ and \mu^- are positive measures on the \sigma-algebra. The Jordan decomposition theorem states that \mu = \mu^+ - \mu^-, where \mu^+ and \mu^- are mutually singular positive measures, and the measure is given by |\mu|(E) = \mu^+(E) + \mu^-(E) for each measurable E. This decomposition holds for any \mu on a (X, \mathcal{A}), and |\mu| is the minimal positive measure dominating \mu in the sense that it extends the behavior of \mu. A key property is the mutual singularity of \mu^+ and \mu^-: there exist disjoint measurable sets A and B with X = A \cup B, \mu^+(B) = 0, and \mu^-(A) = 0, specifically taking A = P and B = N. This ensures the supports of \mu^+ and \mu^- are disjoint up to null sets. The decomposition is unique in the following sense: if \mu = \lambda - \nu for positive measures \lambda and \nu that are mutually singular, then \lambda = \mu^+ and \nu = \mu^-. This uniqueness stems from the fact that any two Hahn decompositions differ only by null sets, leading to the same \mu^+ and \mu^-. Moreover, for any such pair \lambda, \nu, \lambda(E) \geq \mu^+(E) and \nu(E) \geq \mu^-(E) for all E. As an illustrative example, consider the signed measure \mu on the power set of the natural numbers \mathbb{N}, defined by \mu(E) = \sum_{n \in E} \frac{(-1)^n}{n} for E \subseteq \mathbb{N}. Here, the positive set P is the set of even natural numbers (where terms are positive) and N is the set of natural numbers (where terms are negative). The positive part is then \mu^+(E) = \sum_{\substack{n \in E \\ n \text{ even}}} \frac{1}{n}, and the negative part is \mu^-(E) = \sum_{\substack{n \in E \\ n \text{ odd}}} \frac{1}{n}, both of which are positive measures supported on . The total variation is |\mu|(E) = \sum_{n \in E} \frac{1}{n}. This construction follows directly from the definitions using the Hahn sets.

Total Variation Measure

The total variation measure of a signed measure \mu on a measurable space (X, \mathcal{M}) is defined for each E \in \mathcal{M} by |\mu|(E) = \sup\left\{ \sum_{i=1}^\infty |\mu(E_i)| : \{E_i\}_{i=1}^\infty \text{ is a countable partition of } E \right\}, where the supremum is taken over all countable partitions of E into measurable sets. This definition captures the total oscillatory mass of \mu over E. From the Jordan decomposition \mu = \mu^+ - \mu^-, it follows that |\mu|(E) = \mu^+(E) + \mu^-(E), where \mu^+ and \mu^- are the positive and negative parts of \mu. The |\mu| is a positive measure on (X, \mathcal{M}). If \mu is a finite (i.e., \mu(X) > -\infty and \mu(X) < \infty), then |\mu| is also finite. Moreover, |\mu| is the minimal positive measure dominating \mu in the sense that -\,|\mu| \leq \mu \leq |\mu| and any other positive measure \nu satisfying this inequality must fulfill |\mu| \leq \nu. For a finite signed measure \mu absolutely continuous with respect to a positive measure \lambda (so \mu(E) = \int_E f \, d\lambda for some integrable f), the total variation computes as |\mu|(X) = \int_X |f| \, d\lambda, which aligns with the integral form \int |d\mu|. Consider the example on \mathbb{R} with Lebesgue measure m, where \mu(E) = \int_E x \, dm(x) for Borel sets E. Here, |\mu|(E) = \int_E |x| \, dm(x), with the positive and negative supports splitting at $0. The quantity \|\mu\| = |\mu|(X) defines the total variation norm on the space of signed measures, turning it into a Banach space.

Lebesgue Decomposition Theorem

The Lebesgue decomposition theorem provides a canonical way to decompose a signed measure with respect to a reference positive measure, extending the ideas from the Hahn decomposition by incorporating absolute continuity. Specifically, let \lambda be a positive \sigma-finite measure on a measurable space (X, \mathcal{M}), and let \mu be a signed \sigma-finite measure. Then there exist unique (up to \lambda-null sets) signed measures \mu_{ac} and \mu_s such that \mu = \mu_{ac} + \mu_s, where \mu_{ac} \ll \lambda and \mu_s \perp \lambda. The connection to the Hahn decomposition arises in constructing the singular part \mu_s, which is supported on a set where \lambda vanishes. To identify this, consider the signed measure \mu - \epsilon \lambda for small \epsilon > 0; applying the to this measure yields positive and negative sets, and taking an intersection over \epsilon produces a set S \subset X such that \mu_s(E) = \mu(E \cap S) for measurable E, with \lambda(S) = 0 ensuring \mu_s \perp \lambda. Meanwhile, the absolutely continuous part \mu_{ac} admits a Radon-Nikodym derivative: there exists a \lambda-integrable function f: X \to \mathbb{R} such that \mu_{ac}(E) = \int_E f \, d\lambda for all measurable E \subset X, and f = \frac{d\mu_{ac}}{d\lambda} almost everywhere with respect to \lambda. A concrete example illustrates this decomposition on the unit interval [0,1] equipped with \lambda. Consider the signed measure \mu(E) = \int_E g \, d\lambda + c \cdot \delta_0(E), where g \in L^1([0,1], \lambda) is a real-valued integrable function and \delta_0 is the at $0 scaled by a constant c \in \mathbb{R}. Here, \mu_{ac}(E) = \int_E g \, d\lambda is absolutely continuous with respect to \lambda, while \mu_s = c \cdot \delta_0 is singular since \lambda(\{0\}) = 0 but \mu_s(\{0\}) = c \neq 0 (assuming c \neq 0). The decomposition satisfies several key properties that underscore its utility. Uniqueness holds in the sense that if \mu = \mu_{ac}' + \mu_s' is another such splitting, then \mu_{ac} - \mu_{ac}' and \mu_s - \mu_s' are both absolutely continuous and singular with respect to \lambda, implying they vanish on \lambda-null sets. Additionally, the total variation measure decomposes additively as |\mu| = |\mu_{ac}| + |\mu_s|, where |\mu_{ac}| and |\mu_s| are the total variations of the respective parts.

Historical Context

Origins and Development

The Hahn decomposition theorem emerged from early efforts to rigorize the theory of and set functions during the late 19th and early 20th centuries. Preceding Hahn's contributions, Camille Jordan introduced concepts of content and decomposition in the 1880s as part of pre-measure theory, focusing on and functions of to extend Riemann to more general cases. These ideas were further developed by William Henry Young and others in the early 1900s, who refined decompositions for additive set functions, laying groundwork for handling signed measures without full σ-additivity. Hans Hahn advanced this framework in the 1920s through his work on and , introducing the for signed measures in the context of abstract integration theory. Building directly on Jordan's notions of , Hahn formalized the in his 1921 book Theorie der reellen Funktionen, emphasizing partitions of measurable spaces into positive and negative sets relative to a . His contributions appeared in key publications, including explorations of real functions and set functions that bridged classical analysis with emerging measure-theoretic rigor. The theorem arose amid the broader push to solidify Lebesgue's integration theory, which had introduced signed functionals but lacked systematic decompositions for general measures. Hahn's result provided a foundational tool for this rigorization, enabling precise handling of in integration over arbitrary measurable spaces. By the mid-20th century, integrated the theorem into modern measure theory in his 1950 monograph, standardizing its role in abstract treatments of signed measures and their extensions. A notable aspect of the theorem's development concerns its foundational assumptions: the standard proof relies on , a consequence of the (AC), to guarantee the existence of maximal positive sets. In contrast, 1970s counterexamples by Per Enflo demonstrated failures of certain extension principles akin to Hahn-Banach without AC, highlighting that while some functional analytic results can be salvaged in choice-free settings, the decomposition theorem inherently requires AC for its general form.

Related Results

The Vitali–Hahn–Saks theorem provides an extension of concepts related to the Hahn decomposition theorem to sequences of s. Specifically, if \{\mu_n\} is a sequence of s on a \sigma-algebra such that \mu_n(E) converges for every measurable set E, then the defines a , and the sequence is uniformly countably additive provided the measures are finite on a \sigma-finite space. This result ensures that limits of s behave well, avoiding pathologies in convergence that could arise without the decomposition structure. The Hahn decomposition theorem plays a key role in representation theorems for bounded linear functionals on L^\infty spaces. In particular, it enables the identification of the dual of L^\infty(\mu) with the space of bounded finitely additive signed measures on the , where the into positive and negative parts corresponds to the functional's action via against these components. This relies on the Hahn to construct the Jordan form of the associated , ensuring the functional's linearity and boundedness. In non-\sigma-finite settings, the uniqueness of the Jordan decomposition (derived from the Hahn decomposition) requires additional assumptions like \sigma-finiteness, although the Hahn partition itself is unique modulo null sets. Such cases highlight the necessity of \sigma-finiteness for the standard uniqueness properties modulo null sets. Generalizations of the Hahn decomposition extend to vector measures, where a decomposition into "positive" and "negative" components exists for non-atomic vector measures taking values in finite-dimensional spaces, facilitating the proof of Lyapunov's theorem on the convexity of the range of such measures. In non-commutative settings, analogs appear in operator algebras, such as decompositions of bounded self-adjoint linear maps between C*-algebras into differences of positive linear maps, mirroring the signed measure structure. These extensions preserve the core idea of partitioning based on sign or order properties in more abstract lattices. The Hahn decomposition also connects to Lyapunov's theorem, where the decomposition underpins the argument that the range of a non-atomic vector measure in \mathbb{R}^n is convex and compact in the . As a brief , the Lebesgue decomposition theorem builds upon this by separating measures into absolutely continuous and singular parts relative to a reference measure.