What “Rule” Means in the Definition of a Measure

In the sentence

A measure is a rule that assigns a nonnegative size to certain sets,

the word rule means a precise mathematical assignment. It is not necessarily a formula, a physical procedure, or a measuring instrument. It is a way of saying: whenever a set belongs to the allowed collection of measurable sets, the measure gives that set one definite size.

So, if \(X\) is the space we are studying and \(\mathcal{A}\) is the collection of sets we have agreed are measurable, then a measure is a function

\[ \mu:\mathcal{A}\to [0,\infty]. \]

This notation says that \(\mu\) takes each measurable set \(A\in\mathcal{A}\) and assigns it a value \(\mu(A)\), where the value is nonnegative and may be infinite. Here \([0,\infty]\) is the set of nonnegative real numbers together with the possible value \(\infty\). For example, the Lebesgue measure of the whole real line is infinite:

\[ \lambda(\mathbb{R})=\infty. \]

The word “rule” is deliberately broad. Some measures come from familiar formulas, like the length of an interval:

\[ \lambda([a,b])=b-a. \]

Other measures are defined by listing values on a finite space, by using a density function, or by more abstract construction methods such as outer measure and Carathéodory measurability [Halmos 1974; Folland 1999]. What matters is not that the assignment has a simple formula, but that it is well-defined and satisfies the measure axioms.

A Rule Is a Well-Defined Assignment

To say that a measure is a rule means that each measurable set receives one fixed answer. If \(A\) is measurable, then \(\mu(A)\) is not ambiguous. The same set cannot have two different measures under the same measure \(\mu\).

For example, suppose

\[ X=\{1,2,3,4,5,6\} \]

is the set of possible outcomes of rolling a fair die. Let \(\mathcal{A}\) be the collection of all subsets of \(X\). We can define a probability measure \(P\) by

\[ P(A)=\frac{\#A}{6}, \]

where \(\#A\) means the number of elements in \(A\). This is a rule: give me any event \(A\subseteq X\), and I can compute its probability by counting how many outcomes it contains.

For instance, if

\[ A=\{2,4,6\}, \]

then

\[ P(A)=\frac{3}{6}=\frac12. \]

This rule assigns a nonnegative size to the event “the die roll is even.” Because this is a probability measure, the total size of the whole space is \(1\):

\[ P(X)=1. \]

This example also shows why probability is a kind of measure. A probability measure is a measure whose total mass is \(1\), as formalized in Kolmogorov’s axioms of probability [Kolmogorov 1956].

A Rule Is Not Just Any Assignment

There is an important subtlety: not every assignment of numbers to sets is a measure. A measure must obey certain structural rules. The most important one is countable additivity.

If

\[ A_1,A_2,A_3,\dots \]

are pairwise disjoint measurable sets, then a measure must satisfy

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

“Pairwise disjoint” means that no two of the sets overlap. In ordinary language, if a set is split into countably many non-overlapping measurable pieces, then the measure of the whole is the sum of the measures of the pieces.

This is what separates a measure from a casual idea of “size.” A measure is not merely any number attached to a set. It is an assignment compatible with decomposition into countably many pieces.

The measure must also satisfy

\[ \mu(\varnothing)=0, \]

at least in the standard definition. The empty set has size zero. Together with countable additivity and nonnegativity, this gives the basic definition found in standard measure theory texts [Folland 1999; Royden and Fitzpatrick 2010].

So the highlighted sentence is an introductory simplification. More fully, one would say:

\[ \text{A measure on }(X,\mathcal{A})\text{ is a function }\mu:\mathcal{A}\to[0,\infty] \]

such that

\[ \mu(\varnothing)=0 \]

and, for every countable family of pairwise disjoint sets \(A_1,A_2,\dots\in\mathcal{A}\),

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

The parent document’s sentence is therefore correct as a first description, but it is not yet the complete formal definition. The later discussion of sigma-algebras and countable additivity supplies the missing precision.

The Rule Acts Only on “Certain Sets”

The phrase “certain sets” is just as important as “rule.” A measure is not usually defined on every subset of \(X\). Instead, it is defined on a chosen collection \(\mathcal{A}\), called a sigma-algebra.

Thus the measure is not really a function from “all sets” to numbers. It is a function from the measurable sets to nonnegative sizes:

\[ \mu:\mathcal{A}\to[0,\infty]. \]

For a finite die-roll space, it is harmless to let \(\mathcal{A}\) be all subsets of \(X\). But for spaces like \(\mathbb{R}\), if we want a length-like measure that behaves naturally under translations and countable unions, we cannot consistently assign a length to every possible subset of \(\mathbb{R}\), assuming the usual axioms of set theory including the axiom of choice. The classical Vitali construction gives examples of nonmeasurable sets [Folland 1999; Royden and Fitzpatrick 2010].

So the “rule” comes with a domain. Before asking “What is the measure of \(A\)?”, we must know whether \(A\) is in the collection of measurable sets. If \(A\notin\mathcal{A}\), then \(\mu(A)\) is not defined. It is not that the measure is secretly unknown; rather, within that measurable structure, \(A\) is not an allowable input.

This is similar to how a function such as

\[ f(x)=\sqrt{x} \]

is a real-valued function only for \(x\geq 0\). If someone asks for \(f(-1)\) as a real number, the problem is not that we forgot the answer. The input is outside the domain. In the same way, for a measure \(\mu\) defined on \(\mathcal{A}\), only sets in \(\mathcal{A}\) are valid inputs.

Different Rules Can Live on the Same Space

The same underlying set \(X\) can carry many different measures. This is another reason the word “rule” is useful.

For example, on the real line \(\mathbb{R}\), Lebesgue measure \(\lambda\) gives ordinary length. It satisfies

\[ \lambda([0,1])=1. \]

But there is also the counting measure, often denoted \(\#\), defined by

\[ \#(A)= \begin{cases} \text{the number of elements of }A, & \text{if }A\text{ is finite},\\ \infty, & \text{if }A\text{ is infinite}. \end{cases} \]

Under counting measure,

\[ \#([0,1])=\infty, \]

because the interval \([0,1]\) contains infinitely many points.

There is also the Dirac measure at a point \(x_0\), usually written \(\delta_{x_0}\), defined by

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

This measure puts all its mass at a single point. It is important in probability, analysis, and distribution theory.

These examples show that a measure is not “the size” of a set in an absolute sense. It is a chosen rule for assigning size. The interpretation depends on the rule. Lebesgue measure interprets size as length, area, or volume depending on the space. Probability measure interprets size as likelihood. Counting measure interprets size as number of elements. Dirac measure interprets size as whether a special point is included.

A Rule May Be Given by a Formula, but Need Not Be

Sometimes the rule has a simple formula. If \(A\subseteq\{1,2,3,4,5,6\}\), then

\[ P(A)=\frac{\#A}{6} \]

is easy to compute.

Sometimes the rule is given by integration against a density. For example, on \(\mathbb{R}\), if \(f\geq 0\) is an integrable function, one may define

\[ \mu(A)=\int_A f(x)\,dx. \]

Here the measure of \(A\) is the accumulated density over \(A\). If \(f\) represents physical mass density, then \(\mu(A)\) represents the mass contained in the region \(A\). This connects the parent document’s examples of mass, area, volume, and probability.

But in the full construction of Lebesgue measure, the rule is not first introduced as a simple formula for every measurable set. Instead, one constructs an outer measure by covering sets with intervals, then identifies the measurable sets using Carathéodory’s criterion. This is a more abstract rule, but it is still a rule: each Lebesgue measurable set receives one definite nonnegative value [Halmos 1974; Folland 1999].

Why This Word Is Useful

The word “rule” is beginner-friendly, but it points to the formal idea of a function. A measure takes sets as inputs and returns sizes as outputs. The inputs are not points like \(x\); they are sets like \(A\). So, instead of writing something like

\[ f(x), \]

we write

\[ \mu(A). \]

This is one of the first shifts in measure theory. We are no longer only assigning numbers to points. We are assigning numbers to collections of points.

That is why the highlighted sentence is a good first mental model:

A measure is a rule that assigns a nonnegative size to certain sets.

Expanded carefully, it means: choose a space \(X\), choose a sigma-algebra \(\mathcal{A}\) of measurable subsets of \(X\), and define a function \(\mu\) on \(\mathcal{A}\) whose values are nonnegative and whose behavior respects countable unions of disjoint sets. The word “rule” is the informal doorway into this formal structure.

References

Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. 2nd ed., Wiley, 1999.

Halmos, Paul R. Measure Theory. Springer, 1974.

Kolmogorov, A. N. Foundations of the Theory of Probability. 2nd English ed., Chelsea Publishing Company, 1956.

Royden, H. L., and P. M. Fitzpatrick. Real Analysis. 4th ed., Pearson, 2010.