Introduction

In the previous lesson, we introduced singular measures.

We saw that a measure can be singular with respect to Lebesgue measure if it concentrates all of its mass on a set of Lebesgue measure zero.

The simplest example was the Dirac measure:

$$\delta_0$$

which concentrates all mass at a single point.

Today we study a much more sophisticated example:

the Cantor measure.

The Cantor measure is one of the most remarkable objects in all of mathematics because it is:

• continuous
• non-atomic
• singular with respect to Lebesgue measure
• supported on an uncountable set

It shows that singular measures are not merely collections of point masses.

They can be highly distributed while remaining singular.

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Step 1: Constructing the Cantor Set

Everything begins with the interval:

$$[0,1]$$

Remove the middle third:

$$\left(\frac13,\frac23\right)$$

The remaining set is:

$$\left[0,\frac13\right]\cup\left[\frac23,1\right]$$

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Second Step

Remove the middle third from each remaining interval.

We remove:

$$\left(\frac19,\frac29\right)$$

and:

$$\left(\frac79,\frac89\right)$$

The remaining set now contains four intervals.

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Continue Forever

At each stage:

• keep all remaining intervals
• remove the middle third of every interval

After infinitely many steps we obtain the Cantor set:

$$C$$

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Properties of the Cantor Set

The Cantor set possesses astonishing properties.

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Closed

It is a closed subset of:

$$[0,1]$$

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Uncountable

The set contains infinitely many points.

In fact:

$$|C|=|\mathbb R|$$

Its cardinality equals that of the real numbers.

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No Intervals

The Cantor set contains no interval of positive length.

Every interval contains points that were removed.

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Measure Zero

The total length removed is:

$$\frac13+\frac29+\frac4{27}+\frac8{81}+\cdots$$

This geometric series sums to:

$$1$$

Therefore:

$$\lambda(C)=0$$

The Cantor set has Lebesgue measure zero.

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Why This Is Surprising

The Cantor set is:

• uncountable
• infinitely rich
• everywhere complicated

yet:

$$\lambda(C)=0$$

This demonstrates that cardinality and measure are completely different notions.

Measure theory cares about size, not the number of points.

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Step 2: Building the Cantor Measure

Now we place mass on the Cantor set.

Initially:

$$\mu([0,1])=1$$

Think of one unit of probability mass distributed uniformly.

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First Stage

After removing the middle third, place:

$$\frac12$$

of the mass on:

$$\left[0,\frac13\right]$$

and:

$$\frac12$$

on:

$$\left[\frac23,1\right]$$

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Second Stage

Split each interval’s mass equally.

Each of the four remaining intervals receives:

$$\frac14$$

of the total mass.

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Third Stage

Each interval receives:

$$\frac18$$

of the total mass.

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Continue Forever

After infinitely many steps, the limiting measure is called the Cantor measure.

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What Happened to the Removed Intervals?

Every removed interval receives measure zero.

For example:

$$\mu\left(\frac13,\frac23\right)=0$$

Similarly:

$$\mu\left(\frac19,\frac29\right)=0$$

and so on.

All of the probability mass remains trapped inside the Cantor set.

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Support of the Cantor Measure

The support of the Cantor measure is exactly:

$$C$$

the Cantor set.

Since:

$$\lambda(C)=0$$

we immediately obtain:

$$\mu_C \perp \lambda$$

where:

$$\mu_C$$

denotes the Cantor measure.

The Cantor measure is singular with respect to Lebesgue measure.

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Why This Is Different from a Dirac Measure

A Dirac measure concentrates mass at one point.

The Cantor measure does not.

For every point:

$$x\in C$$

we have:

$$\mu_C({x})=0$$

Thus the measure has no atoms.

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Singular Yet Continuous

This is the remarkable feature.

The Cantor measure is:

• singular
• continuous
• non-atomic

Most students initially think singular measures must be point masses.

The Cantor measure proves otherwise.

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Ternary Representation

The Cantor set has an elegant description using base three.

Every point in:

$$C$$

can be written using only digits:

$$0$$

and:

$$2$$

in its ternary expansion.

For example:

$$0.202220022\ldots_3$$

belongs to the Cantor set.

But:

$$0.102_3$$

does not.

The digit:

$$1$$

indicates that the point lies inside a removed middle third.

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Probabilistic Interpretation

Let:

$$X_n$$

be independent random variables such that:

$$P(X_n=0)=\frac12$$

and:

$$P(X_n=2)=\frac12$$

Define:

$$X=\sum_{n=1}^{\infty}\frac{X_n}{3^n}$$

Then the distribution of:

$$X$$

is precisely the Cantor measure.

This interpretation is extremely important in probability theory.

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The Cantor Function

Associated with the Cantor measure is the famous Cantor function.

It is defined by:

$$F(x)=\mu_C([0,x])$$

This is the cumulative distribution function of the Cantor measure.

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Properties of the Cantor Function

The Cantor function is extraordinary.

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Continuous

The function is continuous everywhere.

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Increasing

The function never decreases.

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Constant on Removed Intervals

For example, on:

$$\left(\frac13,\frac23\right)$$

the function is completely flat.

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Maps 0 to 0

$$F(0)=0$$

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Maps 1 to 1

$$F(1)=1$$

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The Devil’s Staircase

The graph of the Cantor function resembles a staircase.

Long flat regions are connected by tiny upward jumps.

Because there are infinitely many levels, the graph became known as:

The Devil’s Staircase

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Derivative of the Cantor Function

A remarkable theorem states:

$$F’(x)=0$$

for Lebesgue almost every:

$$x\in[0,1]$$

Yet:

$$F(1)-F(0)=1$$

This seems impossible at first.

How can a function increase from:

$$0$$

to:

$$1$$

while having derivative zero almost everywhere?

The answer lies in the subtle difference between:

• pointwise behavior
• measure-theoretic behavior

This phenomenon profoundly influenced modern analysis.

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Why Analysts Love the Cantor Measure

The Cantor measure serves as the canonical example of:

• singular continuous measures
• fractal measures
• non-atomic singular measures

Whenever a theorem involves singularity, someone eventually asks:

What happens for the Cantor measure?

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Connection to Hausdorff Dimension

The Cantor set has dimension:

$$\frac{\log 2}{\log 3}$$

This is neither:

$$0$$

nor:

$$1$$

It is fractional.

This observation leads directly to fractal geometry.

Later we will study Hausdorff measure and Hausdorff dimension.

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Connection to Probability

The Cantor measure is a probability measure.

Thus it provides an example of a random variable whose distribution:

• has no density
• has no atoms
• is still continuous

This challenges many intuitions developed from standard probability distributions.

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Connection to Alain Connes

The Cantor set is one of the earliest examples suggesting that ordinary geometric intuition is incomplete.

Classical geometry expects:

• points
• curves
• surfaces

The Cantor set behaves like none of these.

Connes’ noncommutative geometry extends this philosophy much further.

Many modern geometric spaces are too irregular to be described using ordinary points.

The Cantor set is therefore one of the first glimpses of the strange geometric worlds that eventually motivate noncommutative geometry.

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Key Concepts Learned

By the end of this lesson you should understand:

• The Cantor set is uncountable but has measure zero.
• The Cantor measure places all mass on the Cantor set.
• The Cantor measure is singular with respect to Lebesgue measure.
• The Cantor measure is non-atomic.
• The Cantor function is the cumulative distribution function of the Cantor measure.
• The Cantor function is continuous and increasing.
• The Cantor function has derivative zero almost everywhere.
• The Cantor measure is the canonical example of a singular continuous measure.

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Looking Ahead

In the next lesson:

Measure Theory Lesson 20: The Lebesgue Decomposition Theorem

we will combine everything from the past several lessons—absolute continuity, singularity, Radon–Nikodym theory, and signed measures—to prove one of the deepest structural theorems in measure theory:

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

where:

$$\nu_a \ll \mu$$

and:

$$\nu_s \perp \mu$$

This theorem reveals that every measure consists of an absolutely continuous part and a singular part, and it serves as one of the central organizing principles of modern analysis.