What Is a Measure?

Study probability theory and you encounter the words “probability space” and “measurable function”; relearn calculus and you encounter “Lebesgue integral” and “almost everywhere.” At the root of both lies the same concept, the “measure,” but if you study probability and calculus separately it’s hard to notice that the two stand on the same foundation.

This article defines what a measure is, starting from intuition, and organizes how it is used in calculus (the limits of the Riemann integral and the Lebesgue integral) and in probability theory (Kolmogorov’s axiomatization), contrasting the two.

Conclusions

• A measure is a function that abstracts “length, area, volume” and assigns a size to sets. It must, however, satisfy consistency of addition (countable additivity)
• The objects on which a measure can be defined (a $\sigma$-algebra) arise naturally from the structure of “closeness” (open sets, neighborhoods) that metric and topological spaces have, in the form of the Borel $\sigma$-algebra
• Integration can be recast as the operation of partitioning the domain by the values of the integrand and summing with weights given by the measure of those sets
• The Lebesgue integral redefines calculus’s integral in the language of measure, and can handle a broader class of functions than the Riemann integral
• Probability theory is formulated as a measure normalized so that the total measure is exactly $1$ (a probability measure) (Kolmogorov’s axioms, 1933)
• The expectation of a random variable is precisely the Lebesgue integral with respect to the probability measure, and the probability density function is the “density” of the probability measure with respect to Lebesgue measure (the Radon-Nikodym derivative)
• Calculus’s “almost everywhere (a.e.)” and probability theory’s “almost surely (a.s.)” are merely two phrasings of the same concept, “holds except on a set of measure zero”

Prerequisites

• Intended reader: someone who knows the basic terminology of calculus (integration) and probability (random variables, expectation) but wants to systematically understand the common foundation of “measure” behind them
• Aim: not to follow rigorous proofs of measure theory, but to make it click that calculus and probability are both special cases of the same “measure” framework
• Assumed knowledge: an elementary feel for sets, maps, limits, and integration is enough to follow along

Generalizing the Operation of Measuring “Size”

The length of an interval $[a, b]$ is $b - a$, the area of a rectangle is height × width — for naive figures we already know how to measure “size.” What measure theory sets out to do is generalize this operation of “measuring size” to far more complex and abstract sets.

For instance, how should we define the “size” of sets like the following?

• The set of all rationals contained in $[0, 1]$
• The set of all outcomes in which, tossing a coin infinitely many times, the proportion of heads converges to $1/2$
• The set of all points at which some function is not continuous

These are no longer naive figures like intervals or rectangles, but asking about their “size” (whether it’s measure zero, whether the measure is positive, or how much weight it carries as a probability) is meaningful in itself. Measure theory provides a common framework for answering this question.

Definition of a Measure Space

To define a measure rigorously, you first have to decide “which sets we allow ourselves to measure.”

$\sigma$-Algebra: The Collection of Sets We May Measure

A collection $\Sigma$ of subsets of a set $X$ is called a $\sigma$-algebra when it satisfies the following three conditions.

1. $X \in \Sigma$
2. If $A \in \Sigma$, then its complement $A^c \in \Sigma$
3. If $A_1, A_2, \dots \in \Sigma$, then the countable union $\bigcup_{i=1}^{\infty} A_i \in \Sigma$

Sets belonging to $\Sigma$ are called measurable sets. The key point is that it isn’t necessarily possible to define a size for every subset. Under ordinary set theory accepting the axiom of choice (ZFC), it is known that there exist sets on the real line $\mathbb{R}$ to which “length” cannot be consistently assigned (non-measurable sets, such as the Vitali set). That’s exactly why the objects to be measured have to be restricted in advance in the form of a $\sigma$-algebra.

Aside: What Is a “Family”?

First let’s check the word “family” itself. When mathematics says family, it usually just means “a collection of something,” particularly “a set whose elements are sets.” For example, collecting several subsets of the set $X = \{1, 2, 3\}$,

is one example of a family of subsets of $X$. $\Sigma$ is itself a set, but the name “family” is used when you want to emphasize that its elements ($\emptyset$, $\{1\}$, $\{2, 3\}$, $\{1, 2, 3\}$) all share the common property of being “subsets of $X$.” Saying “a set of sets” is confusing, so it’s conventional to call it this — that level of understanding is enough. The word “family” by itself demands no structure yet.

Aside: What Is an “Algebra of Sets”?

Among families, those closed under set operations (the result of an operation lands back in the family) have a special name. A family of subsets $\Sigma$ is called an algebra of sets when it satisfies the following.

• $X \in \Sigma$
• Closed under complements ($A \in \Sigma \Rightarrow A^c \in \Sigma$)
• Closed under finite unions ($A, B \in \Sigma \Rightarrow A \cup B \in \Sigma$)

The Japanese name for this carries the word “additive,” which comes from the requirement that, if you want $\mu(A \cup B) = \mu(A) + \mu(B)$ — the “addition” — to hold for disjoint $A, B$ once a measure is defined, then as a precondition $A \cup B$ itself must also be a set we may measure (an element of $\Sigma$). Closure under unions is exactly the foundation for “being able to add up sizes.”

The $\sigma$-algebra treated in the main text extends this algebra-of-sets condition from “finite unions” to “countably infinite unions.” The symbol $\sigma$ derives from the German Summe, meaning “sum,” and is used as conventional notation indicating that “operations up to countably infinite ones are allowed.” With an ordinary algebra of sets, closed only under finite unions, there’s no guarantee that $\bigcup_{i=1}^{\infty} A_i$ — the union of countably infinitely many sets — lies in $\Sigma$, so even trying to define countable additivity as an infinite series, $\mu\left(\bigcup_i A_i\right) = \sum_i \mu(A_i)$, you don’t know whether the left-hand side’s $\bigcup_i A_i$ is a set you may measure. A $\sigma$-algebra fills that hole and is the framework that lets a measure be defined consistently for countably infinite operations too.

Just reading the conditions as prose makes them hard to picture, so here’s something you can actually manipulate to check what “closed” means. Trying it on the small set $X = \{1, 2, 3\}$, there are only 8 subsets, so you can survey every combination.

Pressing “Example: select only some” selects $\{\emptyset, \{1\}, \{3\}, X\}$. Because the complement of $\{1\}$, namely $\{2, 3\}$, the complement of $\{3\}$, namely $\{1, 2\}$, and the union of $\{1\}$ and $\{3\}$, namely $\{1, 3\}$, are missing, you can see it isn’t an algebra of sets. Pressing “automatically add the missing sets to close it” lets you watch, as an animation, those three actually being added and the family closing.

Aside: Why Can’t Every Subset Be Assigned a Length? (The Vitali Set)

That “there exists a set on $\mathbb{R}$ to which length cannot be consistently assigned” is shown concretely by the following construction (the Vitali set, 1905).

Let the half-open interval from 0 up to but excluding 1 be $I = \{x \in \mathbb{R} : 0 \le x < 1\}$, and define an equivalence relation on this $I$ by “whether the difference is rational.”

This equivalence relation partitions $I$ into equivalence classes consisting of countably many elements (each class has only countably many members because $\mathbb{Q}$ itself is countable). Using the axiom of choice, pick one representative point from each equivalence class, and let $V$ (the Vitali set) be the set collecting them.

For each rational $q$ contained in $I$, consider the set $V_q = (V + q) \bmod 1$ obtained by translating $V$ by $q$ (modulo $1$). Then the following two things hold.

• The $V_q$ are mutually disjoint: if the same element were in both $V_q$ and $V_{q'}$ ($q \ne q'$), two points in $V$ would be duplicate representatives of the same equivalence class, contradicting that one representative was chosen per class
• Every point of $I$ is contained in some $V_q$: the representative of the equivalence class that point belongs to is in $V$, and shifting by a rational gets you back to that point

In other words, combining all countably many $V_q$ gives all of $I$, and they are mutually disjoint.

Now suppose a length $\lambda(V) = c$ could be consistently defined for $V$. Length shouldn’t change under translation (translation invariance), so $\lambda(V_q) = c$ holds for every $q$. The length of a countable union of disjoint sets should be the sum of the individual lengths (countable additivity), so

should hold. But the right-hand side can only be $0$ if $c = 0$, or $\infty$ if $c > 0$. Meanwhile, the left-hand side should be $\lambda(I) = 1$. Since neither $0 = 1$ nor $\infty = 1$ is possible, this is a contradiction.

Trying to assign a length to $V$ while keeping both of the properties everyone expects of length — “length doesn’t change under translation” and “the lengths of countably many disjoint sets can be added” — necessarily produces this contradiction. So $V$ has no choice but to be non-measurable, and it must be excluded in advance from the $\sigma$-algebra (the collection of measurable sets).

Definition of a Measure

A function $\mu: \Sigma \to [0, \infty]$ on a $\sigma$-algebra $\Sigma$ is called a measure when it satisfies the following two conditions.

1. $\mu(\emptyset) = 0$
2. Countable additivity ($\sigma$-additivity): for a sequence of mutually disjoint sets $A_1, A_2, \dots \in \Sigma$,

The triple $(X, \Sigma, \mu)$ is called a measure space. Countable additivity is an intuitively very natural requirement: “adding up the sizes of separate pieces agrees with the size of the whole.” But requiring it even for countably infinitely many sets is the core of what makes measure theory work as a rigorous theory.

Here’s something you can manipulate to check why this “mutually disjoint” condition is needed, and what it means to extend addition to “countably infinitely many.”

In the “finite case” tab you can confirm by area that $\mu(A \cup B) = \mu(A) + \mu(B)$ holds only when A and B don’t overlap (are disjoint), and that with an overlap it’s off by exactly the doubly counted portion. In the “countably infinite case” tab, using the classic example of splitting an interval in half, then in half again, and so on infinitely, you can confirm that the right-hand side of countable additivity, $\sum_{i=1}^{\infty} \mu(A_i)$, is always under 1 at any finite stage but converges to exactly 1 when all infinitely many are combined.

Neither “length” nor “probability” appears in this definition itself. A measure space is purely an abstract framework, and as we’ll see below, what you plug into that framework determines whether you get calculus or probability theory.

Topological and Metric Spaces and Measure: How a Measure Arises from the Structure of “Closeness”

The definition of a measure space is itself abstract and doesn’t tell you which $\sigma$-algebra to use concretely. In fact, the $\sigma$-algebra we ordinarily use on the real line $\mathbb{R}$ is built naturally from the structure of “closeness” that metric and topological spaces have.

Topological Spaces: Treating “Closeness” Without Using Distance

A topological space is a framework for treating “closeness” and “continuity” without using distance. A family $\mathcal{O}$ of subsets of a set $X$ is called a topology on $X$ when it satisfies the following three conditions, and the elements of $\mathcal{O}$ are called open sets.

1. $\emptyset, X \in \mathcal{O}$
2. The union of any number of open sets is open (closed under arbitrary unions)
3. The intersection of finitely many open sets is open (closed under finite intersections)

A set $N$ that contains an open set $U$ containing the point $x$ (that is, a set $N$ for which there exists an open set $U$ with $x \in U \subseteq N$) is called a neighborhood of $x$. It’s good to picture a neighborhood as “a region around that point with some extent to it.” In a topological space, convergence and continuity are all defined using neighborhoods. For example, a sequence $x_n$ converges to $x$ when, whichever neighborhood of $x$ you pick, all sufficiently late $x_n$ lie in that neighborhood. Being able to talk about “closeness” without using distance as a number at all is the strength of topological spaces.

Metric Spaces: Closeness as a Number

A metric space is a more concrete framework than a topological space. When a function $d: X \times X \to \mathbb{R}_{\ge 0}$ on a set $X$ satisfies the following, $d$ is called a metric and $(X, d)$ a metric space.

1. $d(x, y) = 0 \iff x = y$
2. $d(x, y) = d(y, x)$ (symmetry)
3. $d(x, z) \le d(x, y) + d(y, z)$ (triangle inequality)

In a metric space, taking the open ball of radius $r$ centered at $x$, $B(x, r) = \{y \in X : d(x, y) < r\}$, as the basic unit, and defining open sets as “sets writable as an (arbitrary) union of open balls,” gives you a topological space naturally. In other words, a metric space is always also a topological space. The topology induced by the familiar metric $d(x, y) = |x - y|$ on the real line $\mathbb{R}$ is exactly the topology we normally picture as “the collection of open intervals.”

The Borel $\sigma$-Algebra: Building the Measurable Sets from the Open Sets

A topology (a family of open sets) is closed under arbitrary unions and finite intersections, but not under complements. So a topology by itself is not a $\sigma$-algebra. Hence we consider the smallest $\sigma$-algebra containing all open sets. This is called the Borel $\sigma$-algebra, written $\mathcal{B}(X)$. Since it’s “the smallest collection containing every open set and closed under complements and countable unions,” open sets of course, but also closed sets (complements of open sets), countable intersections of open sets, countable unions of closed sets, and so on, are all Borel sets.

For the real line $\mathbb{R}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ generated from the topology of open intervals induced by the metric $d(x, y) = |x - y|$ is the standard starting point when considering Lebesgue measure. In practice it’s common to use the Lebesgue measurable sets (the completion of the Borel $\sigma$-algebra), which widen the measurable objects beyond Borel sets to include “every subset of a set of measure zero,” but the difference between the two is confined to details around measure-zero sets, and for practical purposes it’s fine to think “anything constructible from open and closed sets is measurable.”

To summarize, the flow metric space $\to$ topological space (the structure of open sets induced by the metric) $\to$ Borel $\sigma$-algebra (the collection of measurable sets generated from the open sets) is how “the collection of sets we may measure” arises naturally from the structure of “closeness.”

A measure space itself can be defined without any metric or topological structure (the definition of a measure space presupposes no topology or metric whatsoever). But most of the measures we use day to day, like Lebesgue measure on $\mathbb{R}$, are built by taking in the structure of “closeness” that metric and topological spaces have, in the form of the Borel $\sigma$-algebra.

Calculus and Measure: Lebesgue Measure and the Lebesgue Integral

First let’s apply the measure space framework to the real line $\mathbb{R}$ and look at measure as a generalization of “length.”

Lebesgue Measure: Generalizing “Length”

Taking the length of the interval $(a, b)$ to be $b - a$ is a natural starting point. For any set $A \subseteq \mathbb{R}$, define the outer measure as the infimum of the sums of lengths of countably many open intervals covering $A$.

Re-selecting the measurable sets so that this outer measure satisfies countable additivity gives the Lebesgue measurable sets, and the measure on them is Lebesgue measure. For intervals it gives $\mu((a, b)) = b - a$, matching the naive intuition of “length.” The Lebesgue measurable sets amount to the completion of the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ seen in the previous section (the $\sigma$-algebra generated from the topology of open intervals), with subsets of measure-zero sets added.

The Limits of the Riemann Integral

The integral you learn first in calculus (the Riemann integral) is defined as the limit of Riemann sums: finely partition the domain, multiply the function value on each small interval by the interval’s width, and sum.

This method is naive and easy to handle, but it doesn’t work for functions like the following. Consider the Dirichlet function

on $[0, 1]$. However finely you partition the domain, every small interval necessarily contains both rationals and irrationals, so the upper and lower Riemann sums are always $1$ and $0$ respectively, and the two never agree. In other words, this function is not Riemann integrable.

The Lebesgue Integral: Partitioning the Range

The Lebesgue integral inverts the idea. Rather than partitioning the domain, it partitions the range, and weights by the measure of the set on the domain side taking each value.

First, for a simple function (a function taking only finitely many values) $\varphi = \sum_{i=1}^{n} c_i \, \mathbb{1}_{A_i}$ (where $A_i$ are measurable sets and $\mathbb{1}_{A_i}$ is the indicator function), define the integral as

and define the integral of a general function as an approximation by such simple functions.

Integrating the Dirichlet function along this definition, we have $f = 1 \cdot \mathbb{1}_{\mathbb{Q} \cap [0,1]} + 0 \cdot \mathbb{1}_{[0,1] \setminus \mathbb{Q}}$, and since all rationals contained in $[0, 1]$ form a countable set, the Lebesgue measure is $\mu(\mathbb{Q} \cap [0, 1]) = 0$ (by countable additivity, adding countably many sets of measure zero, one point each, stays 0). Therefore

and the Lebesgue integral computes consistently. The problem with the Riemann integral was that “however you slice the domain side, the value doesn’t settle”; the Lebesgue integral avoids this by “measuring the size of the domain-side set for each value.”

The Relationship Between the Riemann and Lebesgue Integrals

For bounded functions on a bounded interval, the following relations hold.

• If Riemann integrable, then Lebesgue integrable, and the values agree
• The converse does not hold (the Dirichlet function is the counterexample)
• Lebesgue’s criterion for integrability: a bounded function $f$ is Riemann integrable on $[a, b]$ if and only if the set of all its discontinuity points has Lebesgue measure zero

In other words, the Lebesgue integral isn’t merely an alternative method for the Riemann integral; it’s an integral extended so that the objects measured can be handled while ignoring differences on sets of measure zero.

Comparing by Animation: Partitioning the Domain vs. Partitioning the Range

The difference between “partitioning in the domain direction (Riemann)” and “partitioning in the range direction (Lebesgue)” for the same function is easier to grasp by actually moving it than by explaining in words. Switch the partitioning method with the tabs and change the number of partitions $N$ with the slider to see how the color-coded regions and the approximate value change.

As you increase N, the approximate value converges to the same value under both the Riemann and Lebesgue partitions. This very fact — “the way of partitioning differs, yet they converge to the same value” — expresses that the two are essentially just computing the same integral by different methods.

Almost Everywhere (a.e.)

In the Lebesgue integral framework, the fact that “what happens on a set of measure zero doesn’t affect the value of the integral” is used constantly. From this fact, “a property holds except on a set of measure zero” is expressed as holding almost everywhere (a.e.). For example, the Dirichlet function is $0$ almost everywhere (the exception is only the rationals, a set of measure zero).

Probability and Measure: Kolmogorov’s Axioms

Next, we use the same measure space framework, this time to treat “uncertainty.”

The Probability Space $(\Omega, \mathcal{F}, P)$

In 1933, Andrey Kolmogorov axiomatized probability theory in the language of measure theory. A probability space is defined as the following triple.

• $\Omega$: the sample space. The set of all possible outcomes
• $\mathcal{F}$: a $\sigma$-algebra on $\Omega$. The collection of “events” whose probability we may measure
• $P$: a measure on $\mathcal{F}$ satisfying the following (a probability measure)

In other words, a probability measure is nothing other than a measure normalized so that the total size is exactly 1. The requirement of countable additivity carries over unchanged. For mutually exclusive events $A_1, A_2, \dots$, that

holds is exactly the rule “probabilities of mutually exclusive events can be added” that you learn first in a probability textbook. That this comes from the same axiom as the countable additivity of Lebesgue measure becomes visible once you go through measure theory.

A Random Variable Is a Measurable Function

A random variable $X$ is intuitively “something that assigns a number to the outcome of a trial,” but measure-theoretically it is defined as a measurable function.

The condition of “being a measurable function” is merely the requirement that the set “$X$ is at most some value” is properly defined as an event whose probability we may measure (an element of $\mathcal{F}$). This has exactly the same shape as the definition of a measurable function in calculus (the preimage of an interval in the range is a measurable set).

Expectation Is a Lebesgue Integral

The expectation of a random variable $X$ is defined as the Lebesgue integral with respect to the probability measure $P$.

For a discrete random variable, this integral coincides with the familiar sum $E[X] = \sum_i x_i \, P(X = x_i)$. For a continuous random variable, it coincides with

using the density function $f(x)$. Both follow exactly the Lebesgue integral’s construction of “defining the integral of a general function as the limit of integrals of simple functions.”

The Probability Density Function and the Radon-Nikodym Derivative

The term “probability density function” can be understood as a relationship between Lebesgue measure and a probability measure. When a probability measure $P$ is absolutely continuous with respect to Lebesgue measure $\lambda$ (it assigns probability 0 to sets of Lebesgue measure 0), the Radon-Nikodym theorem gives a function $f$ such that

This $f$ is the probability density function. In other words, a density function is the ratio obtained by "dividing" the probability measure by Lebesgue measure, and it plays the role of a bridge connecting the language of calculus with the language of probability theory.

The uniform distribution is the example where this relation is most directly visible. The uniform distribution on the interval $[0, 1]$ is Lebesgue measure itself restricted to $[0, 1]$, and the probability measure and the length measure coincide completely.

Almost Surely (a.s.)

In probability theory, “a property holds except on events of measure zero (probability zero)” is expressed as holding almost surely (a.s.). Measure-theoretically this is exactly the same concept as calculus’s “almost everywhere (a.e.).” Only the name differs by field; the idea that “exceptions may be pushed into a set of measure zero” is shared.

Seen in a Correspondence Table: Lebesgue Measure and Probability Measure

Organizing the correspondences so far into a single table makes it easier to see that calculus and probability theory are both special cases of the same measure space framework.

The only essentially different row in the table is “total measure.” Lebesgue measure becomes infinite when you measure all of $\mathbb{R}$, whereas a probability measure is always normalized to $1$. Aside from the presence or absence of this normalization, the two receive exactly the same theoretical treatment as measures satisfying the same axiom ($\sigma$-additivity).

The Whole Picture in a Diagram

Placing measure theory at the top, real analysis on the calculus side and probability theory can be positioned as branches applying the same measure-space theory to different objects. Hovering over a node in the diagram (or moving with the tab key if using a keyboard) displays the meaning of each in the explanation area below.

The dashed arrow represents the relationship in which the probability density function connects Lebesgue measure and the probability measure via the Radon-Nikodym derivative. Solid lines represent the concretization relation of “applying general measure theory to a specific object,” and the dots moving along the lines show the direction of that application as an animation.

Caveats

• The measures treated in this article are basically a naive explanation assuming finite or $\sigma$-finite measures, and the rigorous constructions of completion and outer measure found in measure theory textbooks are considerably simplified here
• The Radon-Nikodym theorem is usable only when the probability measure is absolutely continuous with respect to Lebesgue measure. For discrete distributions or singular distributions (such as the Cantor distribution), a density function of this form does not exist
• “Almost everywhere” and “almost surely” both mean that exceptions of measure zero can be ignored. But note that measure zero does not imply the empty set (like all the rationals, a countable set has Lebesgue measure 0 but is not empty)
• The existence of non-measurable sets is bound up with the axiom of choice; keep in mind it’s a result under ZFC (for the relationship between the axiom of choice and non-measurable sets, see also the article “A Bird’s-Eye View of Mathematics”)

Summary

• A measure is a function assigning a “size” to sets, satisfying countable additivity. The triple $(X, \Sigma, \mu)$ is called a measure space
• Lebesgue measure generalizes “length,” and the Lebesgue integral redefines integration using this measure by “partitioning the range and weighting by the measure of the corresponding set in the domain”
• Functions the Riemann integral can’t handle (like the Dirichlet function) can be handled by the Lebesgue integral, which can ignore sets of measure zero
• Probability theory is formulated, via Kolmogorov’s axioms, as a measure normalized so that the total measure is $1$ (a probability measure). A random variable is a measurable function, and expectation is a Lebesgue integral
• The probability density function can be understood as the ratio obtained by “dividing” the probability measure by Lebesgue measure (the Radon-Nikodym derivative), and this connects the language of calculus with that of probability theory
• “Almost everywhere (a.e.)” and “almost surely (a.s.)” are two phrasings of the same idea, that sets of measure zero can be ignored as exceptions

References

• A. N. Kolmogorov, “Foundations of the Theory of Probability”, Chelsea Publishing Company (the original is the 1933 German edition)
• Seizo Ito, Introduction to Lebesgue Integration, Shokabo
• Satoru Igari, Introduction to Real Analysis, Iwanami Shoten
• Walter Rudin, “Real and Complex Analysis”, McGraw-Hill
• Naohisa Funaki, Probability Theory, Asakura Shoten
• Patrick Billingsley, “Probability and Measure”, Wiley
• Fuichi Uchida, Sets and Topology, Shokabo
• G. Vitali, “Sul problema della misura dei gruppi di punti di una retta”, 1905 (the original paper on the Vitali set)