Introduction

In the previous lesson, we studied Hausdorff measure and Hausdorff dimension.

We discovered that sets can possess dimensions such as:

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

which are neither whole numbers nor easily described using classical geometry.

Today we move one step further.

Instead of studying fractal sets, we study measures that live on fractal sets.

Examples include:

• Cantor measure
• Self-similar measures
• Invariant measures of dynamical systems
• Measures arising from random fractals

These objects form the foundation of modern fractal geometry and play an important role in probability, ergodic theory, and eventually noncommutative geometry.

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From Sets to Measures

The Cantor set:

$$C$$

has Hausdorff dimension:

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

But knowing the set alone does not tell us how mass is distributed on it.

There are infinitely many measures supported on:

$$C$$

For example:

• Uniform distribution on the Cantor set
• Measures concentrated near one side
• Random measures

Thus geometry and measure are different concepts.

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Self-Similarity

The central idea of fractal geometry is self-similarity.

A set is self-similar if it looks approximately the same at different scales.

The Cantor set satisfies:

$$C=\frac13 C \cup \left(\frac23+\frac13 C\right)$$

This equation says:

The entire Cantor set is composed of two scaled copies of itself.

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Self-Similar Measures

A measure is self-similar if its mass distribution reproduces itself under scaling.

For the Cantor measure:

Half the mass is placed on:

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

and half the mass is placed on:

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

Each piece then repeats the same pattern.

Thus:

The geometry and the measure are both self-similar.

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Iterated Function Systems

A convenient way to describe self-similar fractals is through an Iterated Function System (IFS).

Suppose:

$$S_1,S_2,\ldots,S_m$$

are contraction maps.

A contraction satisfies:

$$|S(x)-S(y)|\le c|x-y|$$

for some:

$$0<c<1$$

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Example: Cantor System

The Cantor set arises from:

$$S_1(x)=\frac{x}{3}$$

and:

$$S_2(x)=\frac{x}{3}+\frac23$$

Repeatedly applying these maps generates the Cantor set.

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Self-Similar Measure Equation

Assign probabilities:

$$p_1,p_2,\ldots,p_m$$

such that:

$$p_1+\cdots+p_m=1$$

The self-similar measure:

$$\mu$$

satisfies:

$$\mu=\sum_{i=1}^{m}p_i(\mu\circ S_i^{-1})$$

This equation completely determines the measure.

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Cantor Measure Revisited

For the Cantor measure:

$$p_1=p_2=\frac12$$

Therefore:

$$\mu=\frac12(\mu\circ S_1^{-1})+\frac12(\mu\circ S_2^{-1})$$

This compact equation captures the entire infinite construction.

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Why This Is Powerful

Instead of describing infinitely many steps, we describe the measure through a single fixed-point equation.

Many modern fractal measures are defined this way.

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Scaling Laws

Suppose:

$$B(x,r)$$

is a ball centered at:

$$x$$

with radius:

$$r$$

For many fractal measures:

$$\mu(B(x,r))\approx r^d$$

for small:

$$r$$

where:

$$d$$

is a dimension.

This scaling law is one of the most important ideas in fractal geometry.

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Local Dimension

The local dimension at a point:

$$x$$

is defined by:

$$d(x)=\lim_{r\to0}\frac{\log\mu(B(x,r))}{\log r}$$

when the limit exists.

This quantity measures how quickly mass accumulates near:

$$x$$

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Interpretation

If:

$$\mu(B(x,r))\approx r^2$$

then:

$$d(x)=2$$

If:

$$\mu(B(x,r))\approx r^{0.63}$$

then:

$$d(x)=0.63$$

The local dimension acts as a measure-theoretic notion of geometric dimension.

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Example: Lebesgue Measure

In:

$$\mathbb R^n$$

we have:

$$\lambda(B(x,r))\approx r^n$$

Therefore:

$$d(x)=n$$

This agrees with ordinary dimension.

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Example: Cantor Measure

For the Cantor measure:

$$d(x)=\frac{\log 2}{\log 3}$$

for almost every:

$$x$$

The measure “feels” the same dimension as the underlying fractal.

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Fractal Measures and Probability

Suppose:

$$X$$

is distributed according to a Cantor measure.

Then:

$$X$$

has:

• no density
• no atoms
• a continuous distribution

This demonstrates that probability distributions can be far more exotic than normal or exponential distributions.

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Invariant Measures

Fractal measures often arise as invariant measures.

A measure:

$$\mu$$

is invariant under a transformation:

$$T$$

if:

$$\mu(T^{-1}(A))=\mu(A)$$

for every measurable set:

$$A$$

Invariant measures become central in ergodic theory.

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Why Invariant Measures Matter

Invariant measures describe long-term behavior.

They tell us:

• where trajectories spend time
• how mass distributes itself
• what statistical properties emerge

Much of modern dynamical systems theory revolves around invariant measures.

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Example: Doubling Map

Consider:

$$T(x)=2x \pmod 1$$

on:

$$[0,1]$$

Lebesgue measure satisfies:

$$\lambda(T^{-1}(A))=\lambda(A)$$

Thus Lebesgue measure is invariant.

This simple example leads directly into ergodic theory.

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Fractals and Dynamics

Many fractals arise naturally as invariant sets of dynamical systems.

Examples include:

• Cantor sets
• Julia sets
• Attractors
• Strange attractors

Their associated measures often reveal more information than the sets themselves.

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Dimension of Measures

For sets we defined Hausdorff dimension.

Measures possess an analogous concept.

The Hausdorff dimension of a measure:

$$\mu$$

is:

$$\dim_H(\mu)=\inf{\dim_H(E):\mu(E)=1}$$

This is the smallest dimension of a set carrying all the mass.

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Why Measure Dimension Matters

Two measures may live on the same set yet possess different dimensions.

Thus:

• geometry of sets
• geometry of measures

are distinct concepts.

Modern research studies both simultaneously.

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Multifractals

Some measures possess different local dimensions at different points.

Instead of one dimension:

$$d$$

there is a whole spectrum of dimensions.

Such measures are called multifractal.

Multifractal theory is an active research area.

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Applications

Fractal measures appear in:

Physics

• Turbulence
• Phase transitions
• Quantum chaos

Biology

• Blood vessel networks
• Lung structures
• Neural systems

Finance

• Market fluctuations
• Scaling phenomena

Computer Science

• Image compression
• Pattern recognition

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Connection to Geometric Measure Theory

Hausdorff measure studies fractal sets.

Fractal measures study how mass distributes on those sets.

Together they form much of the foundation of geometric measure theory.

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Connection to Probability

Many random processes naturally generate fractal measures.

Examples include:

• Brownian motion
• Branching processes
• Random walks on fractals

These subjects lie at the intersection of probability and geometry.

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Connection to Alain Connes

One of the recurring themes in Connes’ work is:

Geometry should be recovered from measure and spectral information.

Fractal measures are early examples where classical geometric intuition begins to fail.

The underlying space may be highly irregular, yet the measure still encodes rich geometric structure.

This idea becomes increasingly important as we move toward noncommutative geometry, where measures, traces, and spectral data often replace ordinary geometric descriptions.

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Key Concepts Learned

By the end of this lesson you should understand:

• Self-similar measures reproduce themselves under scaling.
• The Cantor measure is a self-similar measure.
• Iterated Function Systems generate many fractal measures.
• Local dimension is defined by:

$$d(x)=\lim_{r\to0}\frac{\log\mu(B(x,r))}{\log r}$$

• Measures can possess fractional dimensions.
• Invariant measures connect fractal geometry and dynamical systems.
• Hausdorff dimension can be defined for measures.
• Multifractal measures possess varying local dimensions.

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Looking Ahead

Measure Theory Lesson 35: Geometric Measure Theory — The Big Picture

In the next lesson, we will step back and synthesize everything from Hausdorff measure, fractal dimensions, singular measures, and differentiation theory. This lesson will serve as your introduction to Geometric Measure Theory (GMT), one of the major research areas that connects measure theory, analysis, geometry, minimal surfaces, PDEs, and several mathematical ideas that eventually influence modern geometric and noncommutative frameworks.