Introduction

In the previous lesson, we proved the Radon–Nikodym Theorem.

The theorem stated that if:

$$\nu \ll \mu$$

then there exists a measurable function:

$$f=\frac{d\nu}{d\mu}$$

such that:

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

for every measurable set:

$$A$$

The entire theorem hinges on one condition:

$$\nu \ll \mu$$

This relation is called absolute continuity.

In this lesson, we will study it deeply.

We will see that absolute continuity is much more than a technical assumption. It is a geometric relationship between measures and one of the central concepts connecting measure theory, probability, statistics, and eventually noncommutative geometry.

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Revisiting the Definition

Let:

$$\mu$$

and:

$$\nu$$

be measures on the same measurable space.

We say:

$$\nu$$

is absolutely continuous with respect to:

$$\mu$$

if:

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

for every measurable set:

$$A$$

We write:

$$\nu \ll \mu$$

and read:

ν is absolutely continuous with respect to μ.

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What Does It Really Mean?

Absolute continuity means:

Every null set for μ is also a null set for ν.

In other words:

$$\nu$$

cannot assign mass to a set that:

$$\mu$$

considers invisible.

Thus:

$$\nu$$

is constrained by:

$$\mu$$

and cannot introduce new singular behavior.

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A Visual Interpretation

Imagine:

$$\mu$$

is ordinary area.

Suppose:

$$\mu(A)=0$$

means:

A occupies no area.

Absolute continuity says:

ν is forbidden from placing mass on such a set.

If:

$$A$$

is invisible to:

$$\mu$$

then it must also be invisible to:

$$\nu$$

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Example 1: Constant Density

Let:

$$\mu=\lambda$$

be Lebesgue measure.

Define:

$$\nu(A)=5\lambda(A)$$

Then:

$$\nu \ll \lambda$$

because:

$$\lambda(A)=0$$

immediately implies:

$$\nu(A)=5\lambda(A)=0$$

This is the simplest example.

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Example 2: Variable Density

Define:

$$\nu(A)=\int_A x^2,d\lambda$$

Again:

$$\nu \ll \lambda$$

because whenever:

$$\lambda(A)=0$$

the integral over:

$$A$$

must be zero.

Thus densities automatically produce absolute continuity.

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Example 3: Probability Densities

Suppose:

$$p(x)$$

is a probability density function.

Define:

$$P(A)=\int_A p(x),dx$$

Then:

$$P \ll \lambda$$

This means:

Every probability distribution possessing a density is absolutely continuous with respect to Lebesgue measure.

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A Fundamental Counterexample

Consider the Dirac measure:

$$\delta_0$$

defined by:

$$\delta_0(A)=\begin{cases}
1,&0\in A\
0,&0\notin A
\end{cases}$$

Now observe:

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

but:

$$\delta_0({0})=1$$

Therefore:

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

The Dirac measure concentrates mass on a set of Lebesgue measure zero.

This violates absolute continuity.

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Why the Dirac Measure Matters

The Dirac measure represents an idealized point mass.

It is one of the most important examples in analysis.

Its failure of absolute continuity teaches us that:

Not every measure has a density.

The Radon–Nikodym Theorem applies only when absolute continuity holds.

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Absolute Continuity Is Not Symmetric

Suppose:

$$\nu \ll \mu$$

This does not imply:

$$\mu \ll \nu$$

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Example

Let:

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

Then:

$$\nu \ll \lambda$$

and:

$$\lambda \ll \nu$$

because both have exactly the same null sets.

However:

For:

$$\delta_0$$

and:

$$\lambda$$

we have:

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

and also:

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

since most sets of positive Lebesgue measure receive zero Dirac mass.

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Mutual Absolute Continuity

When:

$$\nu \ll \mu$$

and:

$$\mu \ll \nu$$

we write:

$$\nu \sim \mu$$

and say the measures are equivalent.

Equivalent measures have exactly the same null sets.

They disagree only on how much mass is assigned, not on which sets are negligible.

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Example

Let:

$$d\nu=e^{-x^2}d\lambda$$

on:

$$\mathbb{R}$$

Since:

$$e^{-x^2}>0$$

everywhere,

we obtain:

$$\nu \sim \lambda$$

Both measures share exactly the same null sets.

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Why Equivalence Matters

In probability theory, changing from one equivalent measure to another often corresponds to:

• changing coordinates
• reweighting probabilities
• changing statistical models

Yet the underlying notion of impossibility remains unchanged.

This idea is fundamental in stochastic calculus and Bayesian statistics.

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A Stronger Form of Absolute Continuity

For finite measures, another characterization exists.

A measure:

$$\nu$$

is absolutely continuous with respect to:

$$\mu$$

if:

For every:

$$\varepsilon>0$$

there exists:

$$\delta>0$$

such that:

$$\mu(A)<\delta \implies \nu(A)<\varepsilon$$

This resembles the ε–δ definition from calculus.

The analogy is not accidental.

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Why It Is Called “Absolute Continuity”

Recall ordinary calculus.

A differentiable function satisfies:

$$F(b)-F(a)=\int_a^b F’(x),dx$$

The notion of absolute continuity for functions ensures that tiny intervals produce tiny changes in function values.

For measures, the idea is similar:

Tiny measure under:

$$\mu$$

forces tiny measure under:

$$\nu$$

Hence the terminology.

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Absolute Continuity and Radon–Nikodym

The Radon–Nikodym Theorem tells us:

$$\nu \ll \mu$$

if and only if:

there exists a density:

$$f=\frac{d\nu}{d\mu}$$

such that:

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

Thus absolute continuity and densities are essentially equivalent concepts.

One can think of absolute continuity as:

The property that makes densities possible.

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Absolute Continuity in Probability

Suppose:

$$P$$

and:

$$Q$$

are probability measures.

If:

$$P \ll Q$$

then:

$$P$$

has a density relative to:

$$Q$$

namely:

$$\frac{dP}{dQ}$$

This quantity appears everywhere:

• Bayesian inference
• Likelihood ratios
• Importance sampling
• Stochastic processes
• Information theory

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Likelihood Ratios

In statistics:

$$\frac{dP}{dQ}$$

is often called a likelihood ratio.

Many statistical procedures are built from comparing two probability measures through their Radon–Nikodym derivative.

Thus a large portion of modern statistics ultimately rests upon absolute continuity.

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The Opposite of Absolute Continuity

If absolute continuity means:

Measures share all null sets,

then the opposite situation is:

Measures live on completely different parts of the space.

This leads to the concept of singular measures.

A singular measure has no density with respect to the reference measure.

This is the subject of the next lesson.

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Connection to Alain Connes

The Radon–Nikodym derivative compares measures.

In operator algebras, one studies analogous questions:

How do we compare states?

How do we compare weights?

The resulting theory leads to modular operators and modular automorphisms.

Much of the famous Tomita–Takesaki theory can be viewed as a vast noncommutative generalization of Radon–Nikodym ideas.

Understanding absolute continuity thoroughly is therefore an important conceptual step toward Connes’ work.

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Key Concepts Learned

By the end of this lesson you should understand:

• Absolute continuity means:

$$\nu \ll \mu$$

when:

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

• Absolute continuity is the exact condition required for the Radon–Nikodym Theorem.
• Not every measure is absolutely continuous.
• Dirac measures provide fundamental counterexamples.
• Absolute continuity is not symmetric.
• Equivalent measures satisfy:

$$\nu \sim \mu$$

• Probability densities are Radon–Nikodym derivatives.
• Likelihood ratios arise naturally from absolute continuity.
• Absolute continuity is one half of the measure decomposition theory.

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Looking Ahead

In the next lesson:

Lesson 18: Singular Measures

we will study measures that are as far from absolutely continuous as possible. We will introduce mutual singularity, analyze the Dirac measure and Cantor measure, and begin developing the Lebesgue Decomposition Theorem, one of the deepest structural results in measure theory.