Introduction

In the previous lesson, we introduced signed measures.

A signed measure:

$$\nu$$

behaves like an ordinary measure except that it may take negative values.

We then proved the Hahn Decomposition Theorem, which states that every signed measure splits the underlying space into:

• a positive region
• a negative region

This naturally raises a deeper question:

Can every signed measure be constructed from ordinary positive measures?

The answer is yes.

In fact, every signed measure can be written uniquely as the difference of two positive measures.

This remarkable result is known as the Jordan Decomposition Theorem.

It is one of the most important structural results in measure theory.

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Motivation

Consider a simple example.

Suppose:

$$X={1,2,3}$$

and:

$$\nu({1})=4$$

$$\nu({2})=-3$$

$$\nu({3})=2$$

Notice that:

$$\nu$$

contains both positive and negative contributions.

We might separate them into:

Positive part:

$$\nu^+({1})=4$$

$$\nu^+({2})=0$$

$$\nu^+({3})=2$$

Negative part:

$$\nu^-({1})=0$$

$$\nu^-({2})=3$$

$$\nu^-({3})=0$$

Then:

$$\nu=\nu^+-\nu^-$$

This is exactly the idea behind the Jordan decomposition.

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Statement of the Jordan Decomposition Theorem

Let:

$$\nu$$

be a signed measure.

Then there exist unique positive measures:

$$\nu^+$$

and:

$$\nu^-$$

such that:

$$\nu=\nu^+-\nu^-$$

and:

$$\nu^+$$

and

$$\nu^-$$

are mutually singular.

The measures:

$$\nu^+$$

and:

$$\nu^-$$

are called the positive variation and negative variation of:

$$\nu$$

respectively.

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What Does Mutually Singular Mean?

Two measures:

$$\mu_1$$

and:

$$\mu_2$$

are called mutually singular if there exist disjoint measurable sets:

$$A,B$$

such that:

$$A\cup B=X$$

and:

$$\mu_1(B)=0$$

$$\mu_2(A)=0$$

Intuitively:

• one measure lives entirely on one region
• the other measure lives entirely on a different region

They do not overlap.

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Constructing the Jordan Decomposition

The Hahn Decomposition Theorem provides:

$$P$$

and:

$$N$$

such that:

$$P\cup N=X$$

and:

$$P\cap N=\emptyset$$

where:

• $$P$$ is positive
• $$N$$ is negative

Using these sets we define:

$$\nu^+(A)=\nu(A\cap P)$$

and:

$$\nu^-(A)=-\nu(A\cap N)$$

for every measurable set:

$$A$$

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Why These Are Positive Measures

Since:

$$P$$

is a positive set,

every measurable subset satisfies:

$$\nu(A\cap P)\ge0$$

Therefore:

$$\nu^+$$

is nonnegative.

Similarly:

$$N$$

is a negative set.

Thus:

$$\nu(A\cap N)\le0$$

which implies:

$$-\nu(A\cap N)\ge0$$

Therefore:

$$\nu^-$$

is also nonnegative.

Both are ordinary measures.

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Recovering the Original Signed Measure

Observe that:

$$A=(A\cap P)\cup(A\cap N)$$

and the two pieces are disjoint.

Countable additivity gives:

$$\nu(A)=\nu(A\cap P)+\nu(A\cap N)$$

Substituting the definitions:

$$\nu(A)=\nu^+(A)-\nu^-(A)$$

Thus:

$$\nu=\nu^+-\nu^-$$

as claimed.

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Example

Suppose:

$$X={1,2,3,4}$$

with:

$$\nu({1})=5$$

$$\nu({2})=2$$

$$\nu({3})=-4$$

$$\nu({4})=-1$$

Choose:

$$P={1,2}$$

and:

$$N={3,4}$$

Then:

Positive variation:

$$\nu^+({1})=5$$

$$\nu^+({2})=2$$

Negative variation:

$$\nu^-({3})=4$$

$$\nu^-({4})=1$$

and:

$$\nu=\nu^+-\nu^-$$

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Uniqueness

One of the most remarkable aspects of the theorem is uniqueness.

There is exactly one pair:

$$\nu^+$$

and:

$$\nu^-$$

satisfying:

$$\nu=\nu^+-\nu^-$$

together with mutual singularity.

This means the decomposition is intrinsic to the signed measure itself.

It does not depend on arbitrary choices.

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Total Variation Measure

The Jordan decomposition allows us to define another important object.

The total variation measure is:

$$|\nu|=\nu^++\nu^-$$

This is an ordinary positive measure.

It measures the total amount of positive and negative mass present.

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Example

Suppose:

$$\nu({1})=3$$

and:

$$\nu({2})=-5$$

Then:

$$\nu^+({1})=3$$

$$\nu^-({2})=5$$

Therefore:

$$|\nu|(X)=3+5=8$$

The signed measure itself has total mass:

$$\nu(X)=-2$$

but the total variation equals:

$$8$$

which reflects the total magnitude present.

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Why Total Variation Matters

The total variation measure tells us how large a signed measure really is.

It plays the same role for signed measures that absolute value plays for numbers.

Compare:

$$|3-5|=2$$

versus:

$$|3|+|{-5}|=8$$

The total variation captures the second quantity.

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Signed Measures from Functions

Suppose:

$$f\in L^1(\mu)$$

Define:

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

Then:

$$\nu$$

is a signed measure.

The Jordan decomposition corresponds exactly to splitting:

$$f$$

into its positive and negative parts.

Recall:

$$f^+=\max(f,0)$$

and:

$$f^-=\max(-f,0)$$

Then:

$$f=f^+-f^-$$

and:

$$|f|=f^++f^-$$

Notice the perfect analogy:

Function decomposition:

$$f=f^+-f^-$$

Measure decomposition:

$$\nu=\nu^+-\nu^-$$

Function magnitude:

$$|f|=f^++f^-$$

Measure magnitude:

$$|\nu|=\nu^++\nu^-$$

This is not a coincidence.

It foreshadows the Radon–Nikodym Theorem.

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Why the Jordan Decomposition Is Important

The theorem allows us to reduce questions about signed measures to questions about ordinary positive measures.

Since positive measures are much easier to work with, this decomposition becomes an extremely powerful tool.

Many proofs involving signed measures begin by applying the Jordan decomposition.

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Connection to Functional Analysis

Later we will study linear functionals on spaces such as:

$$L^1$$

and:

$$C(X)$$

Many important representation theorems show that these functionals correspond to signed measures.

The Jordan decomposition then becomes the analogue of decomposing a vector into positive and negative components.

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Connection to Alain Connes

The Jordan decomposition reflects a recurring idea in modern analysis:

Complex objects become understandable once they are decomposed into simpler positive pieces.

This principle appears repeatedly in:

• Functional analysis
• Operator theory
• Spectral theory
• Operator algebras

Connes’ work often studies highly complicated operator-theoretic objects by decomposing them into more manageable components.

The Jordan decomposition is one of the earliest examples of this philosophy.

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Key Concepts Learned

By the end of this lesson you should understand:

• Every signed measure can be written as:

$$\nu=\nu^+-\nu^-$$

• The positive and negative variations are ordinary positive measures.
• The decomposition is unique.
• The measures are mutually singular.
• The total variation measure is:

$$|\nu|=\nu^++\nu^-$$

• The decomposition mirrors the positive-negative decomposition of integrable functions.
• The Jordan decomposition is one of the major structural results in measure theory.

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Looking Ahead

In the next lesson:

Lesson 17: The Radon–Nikodym Theorem

we arrive at one of the deepest and most important results in all of measure theory. The theorem answers the question:

When can one measure be expressed as a density with respect to another measure?

This result forms the mathematical foundation of probability densities, likelihood functions, Bayesian inference, stochastic processes, and much of modern analysis.