Basic Properties of Measures

What is Measure?

Our notion of measure originates in geometry. We seek to assign length, area, or volume to complicated shapes in a way that is consistent with physical intuition. From geometric experience, several fundamental properties emerge:

Nonnegativity
Monotonicity
Euclidean invariance
Decomposability

These principles guide the abstract axioms of measure theory.

Nonnegativity

A measure should never assign a negative value to a set. Geometrically, physical objects have either positive size or zero size. For example, the volume of the unit cube is 1, while the volume of one of its faces is 0.

Later in the theory we will consider signed measures, where negative values are permitted. For instance, in Riemann integration the “area under a curve” may be positive or negative depending on its position relative to the $x$-axis. However, the foundational theory begins with nonnegative measures.

Monotonicity

If one set is contained in another, its measure should not exceed that of the larger set. That is, whenever $A \subset B$, we expect $\mu(A) \leq \mu(B)$.

Monotonicity formalizes the idea that enlarging a set cannot decrease its size. This property will play an important role later when we construct Lebesgue measure via outer approximations.

Euclidean Invariance

In classical geometry, length, area, and volume are invariant under rigid motions such as translations and rotations. A translated or rotated object retains the same size.

While this symmetry is essential in the construction of Lebesgue measure, measure theory itself is far more general. Many measure spaces are not geometric and need not respect Euclidean invariance. Nevertheless, invariance principles often serve as guiding structural constraints.

Decomposability

Perhaps the most fundamental property is decomposability: a complicated set can be partitioned into simpler pieces whose measures are easier to compute. The measure of the whole should equal the sum of the measures of its parts.

The Riemann integral, for example, is defined by decomposing an interval into finitely many subintervals and summing the contributions from each piece. To formalize this idea, we distinguish between finite and countable additivity.

A set function $\mu$ is finitely additive if $\mu(A \sqcup B)=\mu(A)+\mu(B)$ A set function $\mu$ is countably additive if for every sequence of pairwise disjoint sets $(A_n)_{n=1}^\infty$, we have $\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mu(A_n)$

Finite additivity guarantees additivity for a single fixed partition. However, many analytic constructions involve infinite refinement, a sequence of partitions.

Monotone Limits of Sets

A sequence $(A_n)$ is increasing if $A_n \subset A_{n+1}$ for all $n \in \N$. The limit of the increasing sequence, denoted $A_n \nearrow A$, is defined to be

\[A=\bigcup_{n=1}^\infty A_n\]

A sequence $(A_n)$ is decreasing if $A_n \supset A_{n+1}$ for all $n \in \N$. The limit of the decreasing sequence, denoted $A_n \searrow A$, is defined to be

\[A=\bigcap_{n=1}^\infty A_n\]

Increasing sequences represent approximation from below: each stage enlarges the set by adding finer detail. Decreasing sequences represent approximation from above: each stage removes excess portions and refines the set inward.

Suppose $(A_n)$ is increasing such that $A_n \nearrow A$. Using set identities, we may write

\[A=A_1 \sqcup (A_2\setminus A_1) \sqcup (A_3\setminus A_2)\sqcup \dots\]

a countable disjoint union arising from successive refinements. Countable additivity then implies

\[\mu(A)=\sum_{n=1}^\infty \mu(A_n\setminus A_{n-1})=\lim_{n\to\infty}\mu(A_n)\]

Thus, countable additivity guarantees stability under increasing limits, a property known as continuity from below. Finite additivity alone does not ensure this limit identity.

Decreasing sequences require complements. If $(A_n)$ is decreasing and $A_n \searrow A$, then DeMorgan’s law imply $A^c_n \nearrow A^c$ Thus stability under decreasing limits depends on closure under complements together with closure under countable unions. Under appropriate finiteness assumptions, this leads to continuity from above.

The analytic demand that measure behaves consistently under monotone limits leads to requring two structural properties:

closure under countable unions,
closure under complements.

Collections of sets satisfying these properties are precisely $\sigma$-algebras. Countable additivity governs infinite decomposition, and $\sigma$-algebras provide the structural environment in which such decompositions are well defined.

Measurable Spaces and Measures

We now formalize the structures motivated in the previous section.

A measurable space is a pair $(\Omega,\mathcal{F})$ where $\Omega$ is a set and $\mathcal{F}$ is a $\sigma$-algebra of subsets of $\Omega$. The sets in $\mathcal{F}$ are called measurable sets.

The $\sigma$-algebra encodes the structural requirement that the collection of sets be stable under complements and countable unions, and hence under monotone limits.

Let $(\Omega,\mathcal{F})$ be a measurable space. A function $\mu:\mathcal{F}\to[0,\infty]$ is called a measure if:

$\mu(\varnothing)=0$,
For every sequence of pairwise disjoint sets $(A_n)_{n=1}^\infty$ in $\mathcal{F}$, we have

\[\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mu(A_n)\]

The second condition is called countable additivity or $\sigma$-additivity.

A measure space is a triple $(\Omega,\mathcal{F},\mu)$ consisting of a measurable space $(\Omega,\mathcal{F})$ together with a measure $\mu$ on $\mathcal{F}$. We now consider several fundamental examples.

Example: The Dirac Measure

Let $\Omega$ be any set and fix a point $x_0\in\Omega$. The Dirac measure at $x_0$, denoted $\delta_{x_0}$, is defined by

\[\delta_{x_0}(A)=\begin{cases}1 & \text{if } x_0\in A,\\ 0 & \text{if } x_0\notin A.\end{cases}\]

It is straightforward to verify that $\delta_{x_0}$ is a measure on any measurable space $(\Omega,\mathcal{F})$. The Dirac measure concentrates all mass at a single point. It plays a central role in probability theory and in the study of weak convergence of measures.

Example: The Counting Measure

Let $\Omega$ be any set and let $\mathcal{F}=2^\Omega$. The counting measure $\mu$ is defined by

\[\mu(A)=\begin{cases}|A| & \text{if } A \text{ is finite},\\ \infty & \text{if } A \text{ is infinite},\end{cases}\]

where $|A|$ denotes the cardinality of $A$. If $\Omega$ is a finite set, we can renormalize the counting measure by defining

\[p(A)=\frac{|A|}{|\Omega|}\]

This measure coincides with a uniform probability measure on a finite discrete set. We will now record some important properties of measures.

Proposition (Basic Properties of Measures)

Let $(\Omega,\mathcal{F},\mu)$ be a measure space.

(Monotonicity) If $A,B\in\mathcal{F}$ and $A\subset B$, then $\mu(A)\le\mu(B)$.
(Countable Subadditivity) For any sequence $(A_n)$ in $\mathcal{F}$,

\[\mu\left(\bigcup_{n=1}^\infty A_n\right)\le\sum_{n=1}^\infty \mu(A_n)\]

(Continuity from Below) If $(A_n)$ is an increasing sequence in $\mathcal{F}$ and $A_n \nearrow A$, then

\[\mu(A)=\lim \mu(A_n)\]

(Continuity from Above) If $(A_n)$ is a decreasing sequence in $\mathcal{F}$ with $\mu(A_1)<\infty$ and $A_n \searrow A$, then

\[\mu(A)=\lim\mu(A_n)\]

Proof.

The proofs are a straightforward application of countable additivity and set identities.

Special Classes of Measures

Different structural properties of measures lead to important subclasses that play central roles in analysis and probability.

Finite Measures

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. The measure $\mu$ is called finite if $\mu(\Omega)<\infty$. Finite measures behave particularly well under decreasing limits, since continuity from above requires finiteness of the initial set.

$\sigma$-Finite Measures

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. The measure $\mu$ is called $\sigma$-finite if there exists a sequence $(E_n)_{n=1}^\infty$ in $\mathcal{F}$ such that

\[\Omega=\bigcup_{n=1}^\infty E_n \quad \text{and} \quad \mu(E_n)<\infty \text{ for all } n\]

Thus, a $\sigma$-finite measure space can be decomposed into countably many finite pieces. This mild finiteness condition is sufficient for many fundamental results, including uniqueness of extensions, product measure constructions, and the Radon–Nikodym theorem.

Probability Measures

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. The measure $\mu$ is called a probability measure if $\mu(\Omega)=1.$

Probability measures are finite measures and form the foundation of modern probability theory. Later, when studying convergence of measures, probability measures will play a central role.

Atomic and Non-Atomic Measures

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. A measurable set $A$ with $\mu(A)>0$ is called an atom if for every measurable subset $B\subset A$, either $\mu(B)=0$ or $\mu(B)=\mu(A)$. The measure $\mu$ is called atomic if every measurable set of positive measure contains an atom, and non-atomic if it contains no atoms.

Atomic measures behave discretely, while non-atomic measures exhibit a continuous structure. The counting measure and Dirac measure is atomic.

Nonmeasurable Sets

Having defined measures abstractly, it is natural to ask whether we can construct a measure on all subsets of $\mathbb{R}$ that agrees with geometric intuition. More precisely, does there exist a measure $\mu:2^{\mathbb{R}}\to[0,\infty]$ satisfying:

$\mu([a,b])=b-a$ for all intervals
$\mu$ is translation invariant
$\mu$ is countably additive

The answer is no.

Theorem (Vitali)

There is no countably additive, translation-invariant measure defined on all subsets of $\mathbb{R}$ that assigns positive measure to $[0,1]$.

Proof.

By contradiction. Suppose that such a measure $\mu$ exists. Define an equivalence relation on $\mathbb{R}$ by declaring $x\sim y$ if $x-y\in\mathbb{Q}$. The equivalence classes are cosets of $\mathbb{Q}$ in $\mathbb{R}$.

Using the axiom of choice, select exactly one representative from each equivalence class intersecting $[0,1]$. Let $V$ denote this set of representatives. The set $V$ is called a Vitali set.

For each rational number $q\in\mathbb{Q}\cap[-1,1]$, consider the translate

\[V+q=\{v+q:v\in V\}\]

These translates are pairwise disjoint, and their union covers $[0,1]$ up to a bounded region. Translation invariance would force

\[\mu(V+q)=\mu(V)\]

for each such $q$. Let $(q_n)$ be an enumeration of the rationals in $[0,1]$. Countable additivity would then imply

\[\mu([0,1]) \leq \mu\left(\bigcup_{n \in \N} (V+q_n)\right)=\sum_{n=1}^{\infty} \mu(V) \leq \mu([-1,2])\]

We consider two cases:

If $\mu(V)=0$, then the right-hand side equals $0$, contradicting $\mu([0,1])>0$.
If $\mu(V)>0$, then the right-hand side diverges to $\infty$, contradicting the fact that the union is contained in $[-1,2]$, which has finite measure.

In either case, we obtain a contradiction. $\blacksquare$

Consequence

There exist subsets of $\mathbb{R}$ that cannot be assigned a translation-invariant, countably additive notion of length. The failure does not arise from geometry itself, but from the interaction of infinite decomposition and invariance under translation. Countable additivity is too rigid to be compatible with measuring all subsets of $\mathbb{R}$. Thus we are forced to restrict the domain of definition. The natural question becomes:

Which subsets of $\mathbb{R}$ can be assigned length in a manner consistent with countable additivity?

Carathéodory’s construction provides a systematic answer.