A signed measure is a generalization of the concept of a measure that allows the assignment of both non‑negative and non‑positive values to sets in a sigma‑algebra. Formally, let $(X,\mathcal{F})$ be a measurable space, where $X$ is a set and $\mathcal{F}$ is a sigma‑algebra of subsets of $X$. A function $\mu : \mathcal{F} \to [-\infty,\infty]$ is called a signed measure if it satisfies the following properties:
- Null empty set: $\mu(\varnothing)=0$.
- Countable additivity (σ‑additivity): For any countable collection ${A_i}{i=1}^{\infty}$ of pairwise disjoint sets in $\mathcal{F}$, $$ \mu!\left(\bigcup{i=1}^{\infty} A_i\right)=\sum_{i=1}^{\infty} \mu(A_i), $$ where the series converges in the extended real numbers, with the convention that $+\infty - \infty$ is undefined; therefore, a signed measure must be finite on at least one set to avoid indeterminate expressions.
A signed measure may take the value $+\infty$ on some sets and $-\infty$ on others, but it cannot assign both $+\infty$ and $-\infty$ to disjoint subsets of the same set, because this would violate σ‑additivity.
Hahn Decomposition Theorem
For any signed measure $\mu$ on $(X,\mathcal{F})$ there exists a measurable partition $X = P \cup N$ with $P \cap N = \varnothing$ such that:
- $\mu(A) \ge 0$ for every measurable $A \subseteq P$ (the positive set),
- $\mu(A) \le 0$ for every measurable $A \subseteq N$ (the negative set).
The decomposition is not unique, but the sets $P$ and $N$ are uniquely determined up to $\mu$-null sets.
Jordan Decomposition Theorem
Associated with the Hahn decomposition, any signed measure $\mu$ can be expressed uniquely as the difference of two mutually singular finite non‑negative measures: $$ \mu = \mu^{+} - \mu^{-}, $$ where $\mu^{+}(A) = \mu(A \cap P)$ and $\mu^{-}(A) = -\mu(A \cap N)$. The measures $\mu^{+}$ and $\mu^{-}$ are called the positive variation and negative variation of $\mu$, respectively. Their sum $|\mu| = \mu^{+} + \mu^{-}$ is called the total variation of $\mu$. The total variation defines a norm on the space of signed measures: $$ |\mu|_{\text{TV}} = |\mu|(X). $$
Examples
| Example | Description |
|---|---|
| Difference of two measures | If $ |
| u_1$ and $ | |
| u_2$ are (non‑negative) measures on $(X,\mathcal{F})$, then $\mu = | |
| u_1 - | |
| u_2$ is a signed measure. | |
| Lebesgue signed measure | Define $\mu(A) = \lambda(A \cap [0,1]) - \lambda(A \cap (1,2])$, where $\lambda$ denotes Lebesgue measure. Sets wholly contained in $[0,1]$ receive non‑negative values, those in $(1,2]$ receive non‑positive values. |
| Dirac signed measure | For a point $x_0\in X$ and a real constant $c$, the map $\mu(A) = c\mathbf{1}_{{x_0}}(A)$ is a signed measure; it equals $c$ if $x_0\in A$ and $0$ otherwise. |
| Charge (in probability theory) | In probability theory, a signed measure of total mass zero can be used to represent the difference between two probability measures (e.g., $\mu = P - Q$). |
Applications
- Integration with respect to signed measures: The Lebesgue integral can be defined for integrable functions $f$ relative to a signed measure $\mu$ via the Jordan decomposition: $\int f , d\mu = \int f , d\mu^{+} - \int f , d\mu^{-}$, provided at least one of the integrals on the right‑hand side is finite.
- Probability theory: Signed measures model the difference between probability measures, enabling concepts such as total variation distance $ |P - Q|_{\text{TV}} = \frac12 |!P-Q!|(X) $.
- Functional analysis: The space of finite signed measures on a compact Hausdorff space is the dual of the Banach space $C(X)$ of continuous functions (Riesz representation theorem).
- Potential theory and harmonic analysis: Signed measures appear as boundary data for harmonic functions and in the study of distributions of mass with both attractive and repulsive components.
Related Concepts
- Complex measure: A measure taking values in the complex numbers; any complex measure can be expressed as $\mu = \mu_1 + i\mu_2$ with $\mu_1, \mu_2$ signed measures.
- Vector measure: A measure whose values lie in a Banach space; signed measures are the one‑dimensional real case.
- Charge (in the sense of de Groot): Another term for a signed measure, especially in the context of potential theory.
References
- H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th ed., Prentice Hall, 2010.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999.
- W. Rudin, Real and Complex Analysis, 3rd ed., McGraw‑Hill, 1987.
- J. R. Lee, Measure Theory and Integration, Springer, 2021.
