Introduction

In the previous lesson, we studied absolute continuity.

We learned that:

$$\nu \ll \mu$$

means:

$$\mu(A)=0 \implies \nu(A)=0$$

for every measurable set:

$$A$$

This condition is precisely what allows a Radon–Nikodym derivative to exist.

Today we study the opposite extreme.

Instead of asking:

When do two measures see the same negligible sets?

we ask:

When do two measures live on completely different parts of the space?

This leads to one of the most important concepts in measure theory:

Mutual Singularity.

Together, absolute continuity and singularity form the two fundamental ways measures can relate to one another.

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The Basic Idea

Suppose:

$$\mu$$

and:

$$\nu$$

are measures on the same space.

Imagine:

$$\mu$$

lives entirely on one region of the space.

Meanwhile:

$$\nu$$

lives entirely on another region.

Their masses never overlap.

In that situation we say the measures are singular.

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Definition of Mutual Singularity

We say:

$$\mu$$

and:

$$\nu$$

are mutually singular if there exist measurable sets:

$$A,B$$

such that:

$$A\cap B=\emptyset$$

$$A\cup B=X$$

and:

$$\mu(B)=0$$

$$\nu(A)=0$$

We write:

$$\mu \perp \nu$$

and read:

μ is singular with respect to ν.

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Intuition

There exists a partition of the space:

$$X=A\cup B$$

such that:

• all of μ lives inside A
• all of ν lives inside B

The measures completely ignore each other’s territory.

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Visual Picture

Imagine a map.

One measure assigns mass only to the red region.

The other measure assigns mass only to the blue region.

The red and blue regions never overlap.

That is mutual singularity.

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Example 1: Dirac Measure vs Lebesgue Measure

Let:

$$\lambda$$

be Lebesgue measure on:

$$\mathbb{R}$$

and let:

$$\delta_0$$

be the Dirac measure at:

$$0$$

Choose:

$$A=\mathbb{R}\setminus{0}$$

and:

$$B={0}$$

Then:

$$\lambda(B)=\lambda({0})=0$$

and:

$$\delta_0(A)=0$$

Therefore:

$$\delta_0 \perp \lambda$$

This is the most important example of singularity.

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Why This Example Matters

Recall the previous lesson.

We proved:

$$\delta_0 \not\ll \lambda$$

Now we see something stronger.

Not only is:

$$\delta_0$$

not absolutely continuous with respect to:

$$\lambda$$

it actually lives entirely on a set of Lebesgue measure zero.

This is maximal incompatibility.

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Example 2: Two Dirac Measures

Consider:

$$\delta_0$$

and:

$$\delta_1$$

Choose:

$$A={0}$$

and:

$$B=\mathbb{R}\setminus{0}$$

Then:

$$\delta_1(A)=0$$

and:

$$\delta_0(B)=0$$

Therefore:

$$\delta_0 \perp \delta_1$$

Distinct point masses are mutually singular.

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Example 3: A Non-Singular Pair

Let:

$$\nu(A)=2\lambda(A)$$

Then:

$$\nu \ll \lambda$$

and:

$$\lambda \ll \nu$$

These measures share exactly the same support.

They are not singular.

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Support of a Measure

To understand singularity better, we need the idea of support.

Roughly speaking:

The support of a measure is the region where the measure actually lives.

For example:

The support of:

$$\delta_0$$

is:

$${0}$$

The support of:

$$\lambda$$

is:

$$\mathbb{R}$$

Since these supports barely intersect, singularity becomes plausible.

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Singularity vs Absolute Continuity

These concepts represent opposite extremes.

Absolute continuity:

$$\nu \ll \mu$$

means:

ν cannot see anything μ cannot see.

Singularity:

$$\nu \perp \mu$$

means:

ν and μ live on disjoint worlds.

Most of measure decomposition theory consists of understanding how a measure splits into these two components.

---

A Surprising Example: The Cantor Measure

One of the most famous examples in analysis is the Cantor measure.

It arises from the construction of the:

Cantor Set

The Cantor set has remarkable properties:

• uncountable
• contains infinitely many points
• has Lebesgue measure zero

The Cantor measure places all of its mass on this set.

Therefore:

$$\text{Cantor Measure} \perp \lambda$$

even though the support is uncountably infinite.

This surprises many students.

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Why the Cantor Measure Is Important

The Cantor measure demonstrates that singular measures need not be concentrated at points.

A measure may be:

• spread continuously
• have no atoms
• yet still be singular

This phenomenon plays an important role in fractal geometry.

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Atomic and Non-Atomic Measures

An atom is a measurable set:

$$A$$

such that:

$$\mu(A)>0$$

and every measurable subset has either measure:

$$0$$

or:

$$\mu(A)$$

Examples:

• Dirac measures are atomic.
• Lebesgue measure is non-atomic.
• Cantor measure is non-atomic.

Thus singularity and atomicity are completely different concepts.

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Decomposition Philosophy

Suppose:

$$\nu$$

is an arbitrary measure.

Some part may behave like:

$$\lambda$$

and admit a density.

Another part may behave like:

$$\delta_0$$

and be singular.

Can every measure be separated into these pieces?

The answer is yes.

This is one of the deepest results in measure theory.

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The Road to Lebesgue Decomposition

Suppose:

$$\nu$$

and:

$$\mu$$

are measures.

Then:

$$\nu$$

can be split into:

$$\nu=\nu_a+\nu_s$$

where:

$$\nu_a \ll \mu$$

and:

$$\nu_s \perp \mu$$

The first piece has a density.

The second piece is singular.

This is the Lebesgue Decomposition Theorem.

We will prove it soon.

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A Geometric Analogy

Imagine sunlight falling on a surface.

Part of the light is smoothly distributed.

Part is concentrated into sharp spikes.

The smooth component corresponds to absolute continuity.

The spike component corresponds to singularity.

The Lebesgue decomposition separates these effects.

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Why Analysts Care

Singular measures appear everywhere:

• Fractal geometry
• Harmonic analysis
• Dynamical systems
• Probability theory
• Spectral theory

Understanding them is essential for modern analysis.

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Connection to Probability

Probability distributions come in three major types:

Discrete

Example:

$$P(X=0)=1$$

represented by Dirac measures.

---

Continuous

Example:

Normal distributions.

These are absolutely continuous with respect to Lebesgue measure.

---

Singular

Example:

Cantor distributions.

These have no density but are not discrete.

Measure theory explains all three within a single framework.

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Connection to Alain Connes

The distinction between:

$$\ll$$

and:

$$\perp$$

is the first appearance of a recurring theme in modern analysis:

Separate an object into mutually incompatible components.

Later in operator algebras:

• states decompose
• representations decompose
• von Neumann algebras decompose

Connes’ work repeatedly relies on understanding how complicated structures split into fundamentally different pieces.

The absolute continuous/singular decomposition is the prototype of this philosophy.

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Key Concepts Learned

By the end of this lesson you should understand:

• Mutual singularity is written:

$$\mu \perp \nu$$

• Singular measures live on disjoint regions.
• Dirac and Lebesgue measures are singular.
• Distinct Dirac measures are singular.
• Singularity is the opposite extreme of absolute continuity.
• The Cantor measure is singular but non-atomic.
• Every measure can eventually be decomposed into absolutely continuous and singular parts.
• This leads directly to the Lebesgue Decomposition Theorem.

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Looking Ahead

In the next lesson:

Lesson 19: The Cantor Measure and the Cantor Function

we will construct one of the most beautiful and counterintuitive objects in all of analysis. The Cantor measure is continuous, non-atomic, singular with respect to Lebesgue measure, and serves as the canonical example of a singular continuous measure. It occupies a central place in measure theory, fractal geometry, probability, and dynamical systems.