Fact-checked by Grok 2 weeks ago

Measure space

A measure space is a triple (X, \mathcal{A}, \mu) consisting of a set X, a \sigma-algebra \mathcal{A} of subsets of X, and a measure \mu: \mathcal{A} \to [0, \infty] that is countably additive and satisfies \mu(\emptyset) = 0.^[1] In this structure, the \sigma-algebra \mathcal{A} provides the collection of measurable sets, which is closed under complements and countable unions, ensuring that the space supports rigorous definitions of size and integration. The measure \mu generalizes notions like length, area, or volume by assigning a non-negative value (possibly infinite) to each measurable set, with countable additivity guaranteeing that the measure of a countable disjoint union equals the sum of the individual measures.^[1] Measure spaces form the foundation of modern analysis and probability, enabling the construction of integrals over abstract sets beyond Euclidean spaces, such as the Lebesgue integral.^[1] Key properties include monotonicity (if A \subseteq B, then \mu(A) \leq \mu(B)) and countable subadditivity, which follow from the axioms and facilitate theorems like the Carathéodory extension for constructing measures from outer measures.^[1] A special case is the probability space, where \mu(X) = 1, linking measure theory directly to stochastic processes and statistical modeling.^[1] Examples range from the Lebesgue measure on \mathbb{R}^n, which assigns volumes to Borel sets, to Dirac measures concentrating mass at a single point.^[1] Measures may also be \sigma-finite, meaning X is a countable union of sets of finite measure, a condition essential for results like the existence of product measures.^[1]

Definition and Components

Core Elements

A measure space begins with a sample space [X](/page/Sample_space), which is an arbitrary non-empty set serving as the universe of possible outcomes or events under consideration. This set encapsulates all elements relevant to the measurement process, providing the foundational domain upon which more structured components are built. Central to the structure is a \sigma-algebra (or sigma-algebra) \Sigma defined on X, which is a collection of subsets of X known as the measurable sets. Formally, \Sigma must include the empty set \emptyset and the full set X itself, and it is closed under complements: if A \in \Sigma, then X \setminus A \in \Sigma. Additionally, \Sigma is closed under countable unions and countable intersections: for any countable collection \{A_n\}_{n=1}^\infty \subseteq \Sigma, the union \bigcup_{n=1}^\infty A_n \in \Sigma and the intersection \bigcap_{n=1}^\infty A_n \in \Sigma. This closure ensures that the measurable sets form a robust algebra capable of handling limits and infinite operations, establishing \Sigma as the prerequisite framework for applying any measure before quantifying sizes or probabilities.^[2] The measure itself, denoted \mu, is a function \mu: \Sigma \to [0, \infty] that assigns a non-negative extended real number to each measurable set, quantifying its "size" in a consistent manner. A key property is countable additivity: for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subseteq \Sigma (meaning A_i \cap A_j = \emptyset for i \neq j), \mu\left( \bigcup_{n=1}^\infty A_n \right) = \sum_{n=1}^\infty \mu(A_n), with the additional requirement that \mu(\emptyset) = 0. This additivity extends the intuitive notion of length or volume to abstract settings, as seen later in examples like the Lebesgue measure on the real line.^[3]

Formal Definition

A measure space is formally defined as a triple (X, \Sigma, \mu), where X is an arbitrary set, \Sigma is a \sigma-algebra of subsets of X, and \mu: \Sigma \to [0, \infty] is a measure on \Sigma.^[1] The \sigma-algebra \Sigma ensures that the collection of measurable sets is closed under complementation and countable unions, providing a structure suitable for defining limits and integrals. Some texts employ alternative notations, such as \mathcal{M} for the \sigma-algebra, to distinguish it from other collections in the context.^[4] The measure \mu satisfies three key axioms: non-negativity, which is encoded in its codomain of extended non-negative real numbers allowing for infinite values; null-empty set property, \mu(\emptyset) = 0; and countable additivity, which states that for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subset \Sigma,

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

This additivity axiom extends finite additivity to countable collections and handles cases where the sum may diverge to \infty. The allowance for infinite measures accommodates spaces of unbounded size, such as the Lebesgue measure on \mathbb{R}, without restricting to finite total measure.^[1] This formal framework originated in the work of Henri Lebesgue, who introduced measure theory in his 1902 doctoral dissertation to develop a robust theory of integration beyond the Riemann integral.^[5]

Examples

Discrete Cases

In discrete measure spaces, the sample space is a countable set, and the σ-algebra is often the power set of that set, allowing straightforward measurability for all subsets. These spaces highlight the basic principles of measures through simple additive structures on finite or countably infinite domains.^[6] A canonical example is the counting measure on the natural numbers X = \mathbb{N}, equipped with the σ-algebra \Sigma = \mathcal{P}(\mathbb{N}), the power set of \mathbb{N}. The measure \mu is defined such that for any subset A \subseteq \mathbb{N}, \mu(A) = |A| (the cardinality of A) if A is finite, and \mu(A) = \infty otherwise. This construction satisfies countable additivity: for any countable collection of disjoint subsets \{A_n\}_{n=1}^\infty \subseteq \Sigma, \mu\left( \bigcup_{n=1}^\infty A_n \right) = \sum_{n=1}^\infty \mu(A_n), where the sum is understood in the extended reals and may equal infinity. For instance, consider the disjoint singletons \{1\} and \{2\}; then \mu(\{1,2\}) = \mu(\{1\}) + \mu(\{2\}) = 1 + 1 = 2. The counting measure is σ-finite on countable sets like \mathbb{N}, as \mathbb{N} can be covered by the countable union of finite sets \{1\}, \{1,2\}, \{1,2,3\}, \dots, each of finite measure.^[7]^[1]^[8] The Dirac measure provides another discrete example, defined on a countable set X with σ-algebra \Sigma = \mathcal{P}(X). For a fixed point x \in X, the measure \delta_x assigns \delta_x(A) = 1 if x \in A and \delta_x(A) = 0 otherwise, for any A \subseteq X. This measure is finitely additive and, being zero on disjoint sets not containing x, extends to countable additivity; it totals 1 over X, making it a probability measure. The Dirac measure isolates the "mass" at a single point, useful for point evaluations in integration over discrete spaces.^[9]^[1] Finite discrete spaces often model uniform probability distributions, such as a fair coin flip. Here, the sample space is X = \{H, T\} (heads and tails), with \Sigma = \mathcal{P}(X), and the measure \mu defined by \mu(\{H\}) = \mu(\{T\}) = \frac{1}{2}. This setup demonstrates additivity: \mu(X) = \mu(\{H\}) + \mu(\{T\}) = \frac{1}{2} + \frac{1}{2} = 1, and all subsets are measurable with measures adding appropriately for disjoint unions. Such examples underpin discrete probability models where outcomes are equally likely.^[10]^[11]

Continuous Cases

A prominent example of a continuous measure space is the real line \mathbb{R} equipped with the Lebesgue measure \mu, defined on the Lebesgue \sigma-algebra \Sigma, which is the completion of the Borel \sigma-algebra generated by the open sets in \mathbb{R}.^[12] The Lebesgue measure assigns to each closed interval [a, b] the value \mu([a, b]) = b - a, extending this assignment via the Carathéodory criterion to all measurable sets.^[12] This measure is translation-invariant, meaning \mu(E + x) = \mu(E) for any measurable set E \subseteq \mathbb{R} and x \in \mathbb{R}, and non-atomic, as singletons \{x\} have measure zero, ensuring that the measure is diffuse across the continuum rather than concentrated on points.^[12] A specific instance is the unit interval [0, 1] with the restricted Lebesgue measure, where the total measure is \mu([0, 1]) = 1, serving as a foundational space for Lebesgue integration of functions over continuous domains.^[13] More generally, Haar measures provide examples of continuous measures on locally compact topological groups, where the Lebesgue measure on \mathbb{R} under addition exemplifies a left-invariant Haar measure, normalized such that compact sets receive finite measure while preserving invariance under group translations.^[14]

Properties

Fundamental Properties

In a measure space (X, \Sigma, \mu), where \mu is a measure on the \sigma-algebra \Sigma, the axioms of non-negativity and countable additivity imply several fundamental properties that govern the behavior of \mu. Monotonicity states that if A, B \in \Sigma and A \subseteq B, then \mu(A) \leq \mu(B).^[15] To see this, decompose B into the disjoint union B = A \cup (B \setminus A). By countable additivity, \mu(B) = \mu(A) + \mu(B \setminus A). Since \mu(B \setminus A) \geq 0 by non-negativity, it follows that \mu(B) \geq \mu(A).^[15] Countable subadditivity asserts that for any countable collection \{A_n\}_{n=1}^\infty \subseteq \Sigma, \mu\left(\bigcup_{n=1}^\infty A_n\right) \leq \sum_{n=1}^\infty \mu(A_n).^[16] This follows from expressing the union as a disjoint union via the disjointization process and applying countable additivity, combined with monotonicity.^[16] Finite additivity is a direct corollary of countable additivity: for any finite collection of pairwise disjoint sets \{A_k\}_{k=1}^n \subseteq \Sigma, \mu\left(\bigcup_{k=1}^n A_k\right) = \sum_{k=1}^n \mu(A_k).^[1] This holds by setting A_k = \emptyset for k > n in the countable additivity axiom, yielding the finite sum.^[1] Continuity from below provides that if \{A_n\}_{n=1}^\infty \subseteq \Sigma is an increasing sequence (i.e., A_n \uparrow A for some A \in \Sigma), then \mu(A_n) \uparrow \mu(A).^[1] Decompose the union A = \bigcup_{n=1}^\infty A_n into disjoint differences B_1 = A_1 and B_n = A_n \setminus A_{n-1} for n \geq 2; countable additivity then gives \mu(A) = \sum_{n=1}^\infty \mu(B_n) = \lim_{N \to \infty} \sum_{n=1}^N \mu(B_n) = \lim_{N \to \infty} \mu(A_N).^[15] For continuity from above, suppose \{A_n\}_{n=1}^\infty \subseteq \Sigma is decreasing (i.e., A_n \downarrow A for some A \in \Sigma) and \mu(A_1) < \infty. Then \mu(A_n) \downarrow \mu(A).^[1] This follows by applying continuity from below to the complements relative to A_1, using the finite measure assumption to ensure the relevant sets remain of finite measure.^[15]

Completeness and Extensions

A measure space (X, \Sigma, \mu) is called complete if every subset of a null set is measurable, where a null set is any E \in \Sigma with \mu(E) = 0, and such subsets automatically have measure zero as well.^[17] This property ensures that the sigma-algebra \Sigma is closed under taking subsets of negligible sets, preventing "pathological" non-measurable subsets from arising within null sets.^[18] The completion of a measure space (X, \Sigma, \mu) addresses incompleteness by extending the sigma-algebra to include all subsets of null sets while preserving the original measure on \Sigma. Specifically, the completed sigma-algebra \bar{\Sigma} consists of all sets E \subseteq X such that there exist A_1, A_2 \in \Sigma with A_1 \subseteq E \subseteq A_2 and \mu(A_2 \setminus A_1) = 0, and the extended measure \bar{\mu} is defined by \bar{\mu}(E) = \mu(A_1).^[17] Null sets play a central role here, as the completion incorporates their subsets into \bar{\Sigma} without altering \bar{\mu} on the original \Sigma, ensuring \bar{\mu}|_{\Sigma} = \mu.^[18] The resulting space (X, \bar{\Sigma}, \bar{\mu}) is complete and is the smallest such extension containing \Sigma.^[4] For example, the Lebesgue measure on \mathbb{R} is complete, meaning every subset of a Lebesgue null set is Lebesgue measurable.^[16] In contrast, the Borel measure, which is the Lebesgue measure restricted to the Borel sigma-algebra, is not complete, as there exist subsets of Borel null sets (such as certain subsets of the Cantor set) that are not Borel measurable.^[19] The completion of a measure space is unique up to isomorphism, particularly when \mu is sigma-finite, making it the canonical way to achieve completeness without introducing ambiguities in the extension.^[4] This uniqueness follows from the construction as the minimal complete extension, ensuring that any two completions agree on the original sigma-algebra and null sets.^[18]

Important Classes

Probability Spaces

A probability space is a measure space (X, \Sigma, \mu) in which the measure \mu satisfies \mu(X) = 1, and thus \mu is referred to as a probability measure. This structure provides the foundational framework for modeling uncertainty in probability theory through measure-theoretic tools. The concept emerged from Andrey Kolmogorov's 1933 axiomatization, which unified probability with measure theory by treating probabilities as measures on event spaces.^[20] In a probability space, the total measure is finite, specifically equal to 1, ensuring that all subset measures \mu(A) for A \in \Sigma lie between 0 and 1. Measurable sets in \Sigma correspond to events, with \mu(A) denoting the probability P(A) of event A. Integration with respect to the probability measure \mu defines the expectation of a measurable function f: X \to [0, \infty] as \mathbb{E} = \int_X f \, d\mu, linking measure-theoretic integration directly to probabilistic averages. A canonical example is the uniform probability space on the unit interval [0,1], where X = [0,1], \Sigma is the \sigma-algebra of Lebesgue measurable sets, and \mu is the Lebesgue measure restricted to \Sigma, which already satisfies \mu([0,1]) = 1. This space models continuous uniform randomness, such as selecting a point uniformly at random from [0,1].

Sigma-Finite Spaces

A measure space (X, \mathcal{M}, \mu) is called \sigma-finite if the underlying set X can be expressed as a countable union X = \bigcup_{n=1}^\infty X_n, where each X_n is a measurable set with finite measure, i.e., \mu(X_n) < \infty for all n \in \mathbb{N}.^[21] This decomposition allows the space to be approximated by finite-measure subspaces, facilitating the extension of results from finite to infinite settings while preserving key behaviors of countable additivity.^[22] In \sigma-finite spaces, integration and limit operations can be handled by restricting to the finite-measure components and taking limits, which ensures that measurable functions and their integrals align well with the overall structure.^[21] For instance, every finite measure space is \sigma-finite, as X itself serves as the single set of finite measure, and every measure space on a countable set equipped with the counting measure is \sigma-finite, since X decomposes into singletons each of measure 1.^[21] A classic example of a \sigma-finite measure space is the Lebesgue measure on \mathbb{[R](/page/R)}, where \mathbb{[R](/page/R)} = \bigcup_{n=1}^\infty [-n, n] and each interval [-n, n] has finite Lebesgue measure $2n < \infty.^[21] In contrast, the counting measure on an uncountable set, such as the power set of \mathbb{[R](/page/R)}, is not \sigma-finite, because any set of finite measure must be finite (hence countable), and an uncountable union of finite sets cannot cover \mathbb{[R](/page/R)}.^[23] The \sigma-finiteness condition is crucial for many advanced results in measure theory, serving as a prerequisite for theorems like Fubini-Tonelli, which equate iterated and double integrals over product spaces only when both measures are \sigma-finite.^[24] Without it, non-\sigma-finite spaces can exhibit pathologies, such as ill-behaved product measures or failures in the Radon-Nikodym theorem, underscoring its role in ensuring robust theoretical frameworks.^[22]

Standard Measure Spaces

In measure theory, a Borel measure space is constructed from a topological space (X, \tau), where the \sigma-algebra \Sigma is the Borel \sigma-algebra \mathcal{B}(X) generated by the open sets in \tau, and the measure \mu: \mathcal{B}(X) \to [0, \infty] is defined on this \sigma-algebra.^[25] Such spaces arise naturally in analysis when extending measures from open or closed sets to the full Borel structure, ensuring compatibility with the topology.^[1] Borel measures on these spaces are particularly useful for integrating continuous functions and studying convergence properties in topological settings.^[26] A key class consists of standard measure spaces, which are measure spaces whose underlying measurable space is a standard Borel space—that is, a measurable space (X, \mathcal{E}) isomorphic to a Polish space (a separable complete metric space) equipped with its Borel \sigma-algebra—endowed with a \sigma-finite measure \mu.^[27]^[28] These spaces play a central role in descriptive set theory and ergodic theory due to their countable generation and separability, allowing uniform treatment of measurable structures across different Polish topologies.^[29] A canonical example is the Euclidean space \mathbb{R}^n with the Borel \sigma-algebra generated by the standard topology and the Lebesgue-Borel measure, which assigns to each Borel set its Lebesgue measure (as defined in continuous cases).^[1] Under suitable conditions, such as on metric spaces, Borel measures exhibit regularity: for every Borel set E, the measure \mu(E) can be approximated from below by the measures of compact subsets of E (inner regularity) and from above by the measures of open supersets of E (outer regularity).^[1] Finite Borel measures on Polish spaces are always regular in this sense. In probability theory, standard measure spaces underpin many stochastic processes, as they admit complete probability measures that facilitate the construction of random variables with desirable measurability properties.^[30]

References

[1]
[PDF] An Introduction to Measure Theory - Terry Tao
Definition 2.3.1 (Probability space). A probability space is a mea- sure space (Ω, F, P) of total measure 1: P(Ω) = 1. The measure P is known as a ...Missing: source | Show results with:source
[2]
[PDF] Chapter 12: Measure Theory and Function Spaces - UC Davis Math
A measure generalizes volume, associating a nonnegative number to a subset of a set. A measure space is a set, a σ-algebra on it, and the measure.Missing: source | Show results with:source
[3]
[PDF] Introduction to Real Analysis Chapter 10 - Christopher Heil
Jan 25, 2020 · We assumed that we had an abstract measure µ in hand. By definition, such a measure is countably additive on an associated σ-algebra.Missing: core | Show results with:core
[4]
[PDF] Brief Notes on Measure Theory - UC Davis Math
If X is a set and A is a σ-algebra on X, then we call (X,A) a measurable space, and elements of A are called measurable sets. 1 Example If X is any nonempty set ...
[5]
Intégrale, Longueur, aire : Lebesgue, Henri Leon, 1875-1941
Jun 25, 2018 · by: Lebesgue, Henri Leon, 1875-1941. Publication date: 1902. Topics: Integrals. Publisher: Milan : Bernandon de C. Rebeschini.
[6]
[PDF] Measures - UC Davis Math
Counting measure is finite if X is finite and σ-finite if X is countable. A useful implication of the countable additivity of a measure is the following.
[7]
Measure Theory Basics - UC Berkeley Statistics
Aug 24, 2023 · Example 1 (Counting measure): If is countable, e.g. , then a natural measure is the counting measure , which simply counts the number of points ...
[8]
[PDF] 2.2 Measures - Christopher Heil
counting measure on countable sets, such as N or Zd. Counting measure on these sets is important because we will eventually see that integration with.
[9]
[PDF] Lecture 2 Measures
Sep 19, 2013 · A triple (S, S, µ) consisting of a non-empty set, a σ-algebra S on it and a measure µ on S is called a measure space. Remark 2.2.Missing: core | Show results with:core
[10]
[PDF] An Introduction to Discrete Probability - UPenn CIS
Oct 31, 2025 · Consider the sample space of 5 coin flips, with the uniform probability measure (every outcome has the same probability 1/32). Then the ...
[11]
[PDF] 1 Discrete Probability - Jeff Erickson
A discrete probability space (Ω, Pr) has a countable sample space Ω and a probability mass function Pr: Ω → R, where Pr[ω] ≥ 0 and X Pr[ω] = 1.
[12]
Lebesgue Measure -- from Wolfram MathWorld
The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets.
[13]
245A, Notes 1: Lebesgue measure | What's new - Terence Tao
Sep 9, 2010 · We will work inside the unit interval {[0,1]} . For each {x \in [0,1] ... ) More generally, one could define Lebesgue measure using the unit ball ...
[14]
Haar Measure -- from Wolfram MathWorld
If the group is Abelian or compact, then this measure is also right invariant and is known as the Haar measure. More formally, let G be a locally compact group.
[15]
[PDF] Measure Theory John K. Hunter - UC Davis Math
Measures are a generalization of volume; the fundamental example is Lebesgue measure on Rn, which we discuss in detail in the next Chapter. Moreover, as.
[16]
[PDF] 2.3 Basic Properties of Measures - Christopher Heil
(a) µ is a measure if and only if it satisfies continuity from below. (b) If µ(X) < ∞, then µ is a measure if and only if it satisfies continuity from above.
[17]
[PDF] Notes on Real Analysis - University of Houston
Definition 1.8.3. If for every E ∈ M with µ(E)=0, F ⊆ E implies F ∈ M, then µ is called a complete measure on M. Theorem 1.8.4 (Completion of measure space).
[18]
[PDF] 1 August 28, 2013 - CMU Math
Definition 8.6 (Completion of Measure Space). Σµ is the completion of Σ with respect to µ if. Σµ = {A ∪ N | A ∈ Σ, N ∈ N} . Definition 8.7. For every A ∈ Σµ, ...
[19]
[PDF] Lebesgue Measure on Rn - UC Davis Math
The Borel σ-algebra B is not complete and is strictly smaller than the Lebesgue σ-algebra L. In fact, one can show that the cardinality of B is equal to the ...
[20]
[PDF] The origins and legacy of Kolmogorov's Grundbegriffe - arXiv
Feb 5, 2018 · Andrei Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung, which set out the axiomatic basis for modern probability theory, appeared in ...
[21]
[PDF] Real Analysis lecture notes for MA 645/646
Next are the concepts of continuity, derivative, and integral. While at least the ideas, if not the formal definition ... Suppose (X, M,µ) is a measure space with ...<|control11|><|separator|>
[22]
[PDF] Class notes, 6211 and 6212 - OSU Math
Apr 22, 2019 · Two landmark discoveries are typically credited in the development of analysis: Calculus (circ. 1665) and Fourier series, introduced by ...
[23]
[PDF] Chapter 17. General Measure Spaces: Their Properties and ...
Apr 15, 2019 · Lebesgue measure on R is a σ-finite measure. The counting measure on an uncountable set is not σ-finite. Note. Many of the properties of ...
[24]
[PDF] 04b. Product measures and Fubini-Tonelli theorem
Nov 7, 2018 · Fubini-Tonelli theorem(s). Let X, µ and Y,ν be measure spaces with corresponding σ-algebras A, B. Assume X and Y are σ-finite. [2.1] Theorem ...
[25]
[PDF] Measures and Measure Spaces
A triple (X, A,µ) is called a measure space where A is a σ-algebra in P(X), and µ is a measure on A. If E ∈ A, then E is called A-measurable, or more commonly ...
[26]
[PDF] On Radon Measure
Dec 3, 2016 · A Radon measure is a Borel measure that is finite on compact sets, outer regular on all Borel sets, and inner regular on open sets.
[27]
[PDF] Some Notes on Standard Borel and Related Spaces - arXiv
A measurable space is a pair (X, E) consisting of a non-empty set X together with a σ-algebra E of subsets of X. If (X, E) and (Y, F) are measurable spaces ...<|control11|><|separator|>
[28]
[PDF] Measure Theory, 2010 - UCLA Logic Center
Definition A set X equipped with a σ-algebra Σ is said to be a standard Borel space if there is some choice of a Polish topology on X which gives rise to Σ as ...<|control11|><|separator|>
[29]
[PDF] Theory of Probability - University of Texas at Austin
Definition 9.13 (Borel spaces) A measurable space (S,S) is said to be a Borel space (or a nice space) if it is isomorphic to a Borel subset of R, i.e., if ...