Mathematics.

generalizations of measure

Signed Measures

Measure Theory75 minDifficulty8 out of 10

You should know: sigma algebras, lebesgue measure, null sets and almost everywhere, borel sets

Overview

A signed measure generalises a positive measure by allowing sets to have negative measure. This is essential for representing differences of measures, for defining charges in electrostatics, and for the Radon–Nikodym theorem. Every signed measure decomposes uniquely (Jordan decomposition) into the difference of two positive measures supported on disjoint sets (Hahn decomposition).

Intuition

A signed measure assigns a real number (possibly negative) to each measurable set. Think of it as representing a charge distribution: some regions carry positive charge, others negative. The Hahn decomposition theorem says you can always find a 'watershed': a set P (positive) and its complement N (negative) such that every sub-set of P has non-negative measure and every sub-set of N has non-positive measure. The Jordan decomposition then writes the signed measure as mu = mu+ - mu-, the difference of the restrictions to P and N.

Formal Definition

Definition

Let (X, M) be a measurable space. A signed measure is a function nu: M -> [-inf, inf] such that: (1) nu(empty) = 0, (2) nu takes at most one of the values +inf, -inf, (3) for any sequence of disjoint measurable sets {E_n}, nu(union E_n) = sum nu(E_n) (with the sum converging absolutely if nu(union E_n) is finite).

ν(∅)=0\nu(\emptyset) = 0
Null empty set
ν ⁣(⨆n=1∞En)=∑n=1∞ν(En)\nu\!\left(\bigsqcup_{n=1}^{\infty} E_n\right) = \sum_{n=1}^{\infty} \nu(E_n)
Countable additivity for signed measures
ν=ν+−ν−(Jordan decomposition)\nu = \nu^+ - \nu^- \quad \text{(Jordan decomposition)}
Jordan decomposition
∣ν∣=ν++ν−(total variation measure)|\nu| = \nu^+ + \nu^- \quad \text{(total variation measure)}
Total variation

Notation

NotationMeaning
ν\nuA signed measure
ν+,  ν−\nu^+,\; \nu^-Positive and negative parts in the Jordan decomposition
∣ν∣|\nu|Total variation measure of nu
ν⊥μ\nu \perp \munu and mu are mutually singular

Theorems

Theorem 1: Hahn Decomposition Theorem
For any signed measure ν on (X,M), there exist disjoint P,N∈M with P∪N=X such that ν(E)≥0 for all E⊆P and ν(E)≤0 for all E⊆N. The pair (P,N) is called a Hahn decomposition of ν.\text{For any signed measure } \nu \text{ on } (X, \mathcal{M}), \text{ there exist disjoint } P, N \in \mathcal{M} \text{ with } P \cup N = X \text{ such that } \nu(E) \geq 0 \text{ for all } E \subseteq P \text{ and } \nu(E) \leq 0 \text{ for all } E \subseteq N. \text{ The pair } (P,N) \text{ is called a Hahn decomposition of } \nu.
Theorem 2: Jordan Decomposition Theorem
There exist unique positive measures ν+ and ν− on (X,M) such that ν=ν+−ν− and ν+⊥ν−.\text{There exist unique positive measures } \nu^+ \text{ and } \nu^- \text{ on } (X,\mathcal{M}) \text{ such that } \nu = \nu^+ - \nu^- \text{ and } \nu^+ \perp \nu^-.
Theorem 3: Total variation
∣ν∣(E)=ν+(E)+ν−(E)=sup⁡ ⁣{∑k∣ν(Ek)∣:{Ek} is a finite partition of E}|\nu|(E) = \nu^+(E) + \nu^-(E) = \sup\!\left\{ \sum_k |\nu(E_k)| : \{E_k\} \text{ is a finite partition of } E \right\}

Worked Examples

  1. This is a signed measure since nu(E) = mu1(E) - mu2(E) is real-valued and countably additive.

  2. One Hahn decomposition: P = {x : d(mu1)/d(mu1+mu2) >= 1/2} (by Radon-Nikodym). More concretely, consider any set where mu1 dominates.

  3. A simpler example: on X = {1,2,3} with M = P(X), let mu1({1})=3, mu1({2})=1, mu1({3})=0 and mu2({1})=1, mu2({2})=2, mu2({3})=1. Then nu({1})=2, nu({2})=-1, nu({3})=-1.

    ν({1})=2,ν({2})=−1,ν({3})=−1\nu(\{1\}) = 2,\quad \nu(\{2\}) = -1,\quad \nu(\{3\}) = -1

  4. Hahn decomposition: P = {1} (positive), N = {2,3} (negative). Check: nu(E) >= 0 for E ⊆ P and nu(E) <= 0 for E ⊆ N.

Answer: P = {1}, N = {2,3}. Jordan: nu+ = 2*delta_1, nu- = delta_2 + delta_3.

Practice Problems

Difficulty 7/10

Let nu be a signed measure and {E_n} a sequence of disjoint measurable sets. Prove that sum |nu(E_n)| < inf (the series converges absolutely) whenever nu(union E_n) is finite.

Difficulty 8/10

Let f in L^1(R, lambda). Define nu(E) = integral_E f dlambda. Show nu is a signed measure and find its Jordan decomposition.

Difficulty 6/10

Two measures mu and nu are mutually singular (nu perp mu) means:

Common Mistakes

Common Mistake

A signed measure can take both +inf and -inf as values.

By definition, a signed measure takes at most one of the values +inf or -inf. This prevents indeterminate forms like inf - inf in the additivity condition.

Common Mistake

The Hahn decomposition is unique.

The Hahn decomposition (P, N) is unique up to nu-null sets: if (P', N') is another Hahn decomposition, then P delta P' is a null set for |nu|.

Common Mistake

Every signed measure is the difference of two arbitrary positive measures.

The Jordan decomposition is the canonical difference of mutually singular positive measures. Any representation nu = mu1 - mu2 must satisfy mu1 >= nu+ and mu2 >= nu- (the Jordan parts are the minimal choice).

Quiz

A signed measure differs from a positive measure in that it:
The Jordan decomposition of a signed measure nu gives:
The total variation |nu| of a signed measure nu = nu+ - nu- is:

Historical Background

Signed measures arose naturally when analysts needed to represent the difference of two measures, for example in studying absolutely continuous functions and their derivatives. The Hahn decomposition theorem was proved by Hans Hahn in 1921, and Jordan decomposition is named after Camille Jordan who developed decomposition ideas for functions of bounded variation.

  1. 1881

    Jordan introduces bounded variation and total variation for functions

    Camille Jordan

  2. 1921

    Hahn proves the Hahn decomposition theorem for signed measures

    Hans Hahn

  3. 1930s

    Signed measures systematically incorporated into abstract measure theory

Summary

  • A signed measure generalises a positive measure by allowing negative values, subject to countable additivity and taking at most one of ±inf.
  • The Hahn decomposition theorem partitions X into a positive set P and negative set N for any signed measure.
  • The Jordan decomposition writes nu = nu+ - nu- uniquely, where nu+ and nu- are mutually singular positive measures.
  • The total variation |nu| = nu+ + nu- is a positive measure capturing the total size of nu.
  • Signed measures are the natural setting for the Radon–Nikodym theorem and for dual spaces of L^1.

References

  1. BookFolland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley. Chapter 3.
  2. BookRudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill. Chapter 6.
  3. WebsiteWikipedia — Signed measure