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Null set

In mathematics, particularly in measure theory, a null set (or set of measure zero) is a measurable set whose measure is zero. This concept is central to Lebesgue measure on the real numbers, where a null set can be covered by a countable collection of intervals with arbitrarily small total length.^[1] Unlike the empty set, which is always a null set, null sets can be non-empty and even uncountable, such as the rational numbers or the Cantor set.^[2] Key properties of null sets include closure under countable unions and subsets: if each set in a countable collection has measure zero, so does their union, and any subset of a null set is also null. These properties make null sets "negligible" in integration and analysis, often ignored without affecting results.^[3] In broader contexts, null sets appear in probability theory as events with probability zero and in topology related to measurability. The notion of null sets originated with Henri Lebesgue's development of measure theory in his 1902 doctoral thesis Intégrale, longueur, aire, providing a rigorous framework for integration beyond Riemann's method.^[4] Today, null sets are fundamental in real analysis, probability, and other fields, underpinning theorems like the almost everywhere convergence of functions and the completeness of measure spaces.

Definition

In Measure Theory

In measure theory, a null set is formally defined within a measure space (X, \Sigma, \mu), where X is a set, \Sigma is a \sigma-algebra of subsets of X, and \mu: \Sigma \to [0, \infty] is a measure. A set N \in \Sigma is a null set if \mu(N) = 0.^[5] This condition applies exclusively to measurable sets, as the measure \mu is only defined on \Sigma. The collection of all null sets in the space, consisting of those measurable sets assigned zero measure, forms the \sigma-ideal of null sets, which are negligible with respect to \mu. The concept of null sets originated in Henri Lebesgue's work on integration and measure around 1900, particularly in his 1902 doctoral dissertation where he introduced outer and inner measures to characterize measurable sets and highlighted the role of sets of measure zero.^[6] It was later generalized by Constantin Carathéodory in 1914, who developed an axiomatic framework for abstract measures on arbitrary spaces, extending the notion beyond the real line to general measure spaces via the Carathéodory measurability criterion.^[7] In this abstract setting, the empty set \emptyset is always null since \mu(\emptyset) = 0, but null sets can be non-empty, distinguishing them from the empty set; for instance, in many measures, singletons or countable sets qualify as null.^[5] For non-measurable sets, the notion of nullity is extended using the outer measure \mu^* induced by \mu, defined on the power set \mathcal{P}(X) as \mu^*(E) = \inf \left\{ \sum_{n=1}^\infty \mu(A_n) : E \subseteq \bigcup_{n=1}^\infty A_n, A_n \in \Sigma \right\} for E \subseteq X. A set E \subseteq X is then considered null in this broader sense if \mu^*(E) = 0, even if E \notin \Sigma.^[5] In incomplete measure spaces, where not all subsets of null sets are necessarily measurable, the completion of the space incorporates such subsets into an enlarged \sigma-algebra, ensuring that all subsets of null sets become measurable and retain measure zero.^[8] This extension preserves the intuitive idea of null sets as negligible while accommodating the limitations of the original \sigma-algebra.^[8]

In Lebesgue Measure

In the framework of measure theory, a null set is one with measure zero under a given measure; the Lebesgue measure provides a specific realization of this concept on the Euclidean spaces \mathbb{R}^n.^[3] A subset N \subseteq \mathbb{R} is Lebesgue null if, for every \varepsilon > 0, it can be covered by a countable collection of open intervals whose total length is less than \varepsilon.^[3] This condition ensures that N has Lebesgue outer measure zero, defined as

m^*(N) = \inf\left\{ \sum_{k=1}^\infty \ell(I_k) \;\middle|\; N \subseteq \bigcup_{k=1}^\infty I_k, \; I_k \text{ open intervals in } \mathbb{R} \right\} = 0,

where \ell(I_k) denotes the length of each interval I_k.^[3] Equivalently, N is null if no positive length can be assigned to it via such coverings.^[9] This definition extends naturally to higher dimensions \mathbb{R}^n by replacing intervals with n-dimensional rectangles or cubes, where the volume replaces length in the covering sum.^[3] The Lebesgue outer measure on \mathbb{R}^n is constructed as the product measure of the one-dimensional Lebesgue measures on each coordinate, leveraging Fubini's theorem to ensure consistency across dimensions; thus, a subset N \subseteq \mathbb{R}^n is null if

m^*(N) = \inf\left\{ \sum_{k=1}^\infty \mathrm{vol}(R_k) \;\middle|\; N \subseteq \bigcup_{k=1}^\infty R_k, \; R_k \text{ rectangles in } \mathbb{R}^n \right\} = 0,

with \mathrm{vol}(R_k) the n-dimensional volume.^[10] Null sets under Lebesgue measure are always measurable, as the Lebesgue \sigma-algebra is complete: any subset of a null set is itself null and hence measurable.^[3] This completeness property distinguishes the Lebesgue measure among common measures on \mathbb{R}^n.

Examples

Countable Null Sets

In the context of Lebesgue measure on \mathbb{R}^n, every countable set possesses measure zero, a fundamental property that underscores the "nullity" of discrete structures in continuous spaces. This holds because the Lebesgue outer measure of a countable set E = \{x_i : i \in \mathbb{N}\} can be made arbitrarily small by covering it with intervals of shrinking lengths. Specifically, for any \varepsilon > 0, enumerate the points and cover each x_i with an open interval I_i of length \varepsilon / 2^i; the total measure of the cover is then \sum_{i=1}^\infty |I_i| = \varepsilon, implying the outer measure \mu^*(E) \leq \varepsilon. Since \varepsilon is arbitrary, \mu^*(E) = 0.^[3] Finite sets represent the simplest case of countable null sets, as the measure of a singleton \{x\} is zero—covered by an interval of length \varepsilon for arbitrary \varepsilon > 0—and finite unions preserve this property via subadditivity of the outer measure. More generally, the countable union of null sets remains null, following from the countable subadditivity of Lebesgue outer measure: if each E_i satisfies \mu^*(E_i) = 0, then \mu^*(\bigcup_{i=1}^\infty E_i) \leq \sum_{i=1}^\infty \mu^*(E_i) = 0.^[3] This closure under countable unions highlights the robustness of null sets in measure theory. Prominent examples include the natural numbers \mathbb{N}, integers \mathbb{Z}, and rational numbers \mathbb{Q}, all of which are countable and thus null in \mathbb{R}. The set \mathbb{Q} is particularly illustrative as a dense countable null set, intersecting every nonempty open interval despite having measure zero, demonstrating that nullity does not preclude topological density. Similarly, the algebraic numbers—the roots of nonzero polynomials with rational coefficients—form a countable set and hence have Lebesgue measure zero in \mathbb{R}, as they are the countable union over degrees n of the roots of polynomials with integer coefficients, each finite in number.

Uncountable Null Sets

The standard middle-thirds Cantor set C \subseteq [0,1] provides a canonical example of an uncountable null set in the context of Lebesgue measure. It is constructed iteratively: begin with the closed interval [0,1]; at the first stage, remove the open middle-third interval (1/3, 2/3), leaving two closed intervals each of length $1/3; at the second stage, remove the open middle-third from each remaining interval, leaving four closed intervals each of length $1/9; continue indefinitely. The resulting Cantor set is the intersection C = \bigcap_{n=1}^\infty C_n, where C_n denotes the union of the $2^n closed intervals of length $3^{-n} at stage n. This set is compact, totally disconnected, and perfect (closed with no isolated points).^[11] To establish that C has Lebesgue measure zero, observe that the total length of the removed intervals sums to 1: at stage n, $2^{n-1} open intervals of length $3^{-n} are removed, yielding a total removed length of \sum_{n=1}^\infty 2^{n-1} / 3^n = 1. Since the Lebesgue measure of [0,1] is 1 and the measure is countably additive, the measure of C is m(C) = 1 - 1 = 0. More directly, for any \varepsilon > 0, the $2^n intervals comprising C_n cover C with total length (2/3)^n, which can be made less than \varepsilon by choosing n sufficiently large, confirming nullity via the definition of Lebesgue measure.^[11] Despite having measure zero, C is uncountable, with cardinality equal to the continuum, |C| = 2^{\aleph_0}. This follows from representing points in C using ternary expansions consisting only of digits 0 and 2 (avoiding 1 to exclude removed intervals), which establishes a bijection with the set of infinite binary sequences via the map sending 0 to 0 and 2 to 1; the latter set has cardinality $2^{\aleph_0}, matching that of the real numbers in [0,1]. The Cantor set is self-similar, with Hausdorff dimension \log 2 / \log 3 \approx 0.6309 < 1, reflecting its fractal structure while underscoring that measure zero does not preclude substantial topological complexity.^[11]^[12] For contrast, constructions analogous to the Cantor set but removing less length at each stage yield "fat" Cantor sets that are uncountable, compact, and nowhere dense yet possess positive Lebesgue measure; however, variants designed to remove sufficient length—such as adjusted middle-interval removals totaling length 1—produce additional uncountable null sets. The Smith-Volterra-Cantor set, for instance, removes middle intervals of length $1/4^n from $2^{n-1} segments at stage n, resulting in total removed length $1/2 and thus positive measure $1/2, serving as a counterexample to the notion that all such iterative null-like constructions must have zero measure. Focus remains on zero-measure cases like the standard Cantor set, which illustrate that uncountability and nullity can coexist without intervals.^[13]^[14]

Properties

Algebraic Properties

In a measure space (X, \mathcal{M}, \mu), the collection \mathcal{N} of null sets—defined as the measurable sets E \in \mathcal{M} with \mu(E) = 0—forms a \sigma-ideal within the \sigma-algebra \mathcal{M}. This structure means \mathcal{N} contains the empty set, is closed under countable unions, and is closed under taking measurable subsets: if N \in \mathcal{N} and M \in \mathcal{M}, then N \cap M \in \mathcal{N}.^[15]^[16] The closure under subsets follows from the monotonicity of measures: if \mu(A) = 0 and B \subseteq A with B \in \mathcal{M}, then \mu(B) \leq \mu(A) = 0, so \mu(B) = 0. Similarly, closure under countable unions is a consequence of countable subadditivity: if \{A_i\}_{i=1}^\infty \subseteq \mathcal{N}, then

\mu\left( \bigcup_{i=1}^\infty A_i \right) \leq \sum_{i=1}^\infty \mu(A_i) = \sum_{i=1}^\infty 0 = 0,

implying \bigcup_{i=1}^\infty A_i \in \mathcal{N}. Finite unions of null sets are a trivial special case, as they can be viewed as countable unions with all but finitely many A_i = \emptyset.^[15]^[16] In finite measure spaces where \mu(X) < \infty—corresponding to bounded domains under measures like Lebesgue measure—the complement of a null set has full measure. For A \in \mathcal{N},

\mu(X \setminus A) = \mu(X) - \mu(A) = \mu(X) - 0 = \mu(X),

by the additivity of measures on disjoint sets. This property highlights the "negligible" nature of null sets relative to the entire space.^[15] In \sigma-finite measure spaces, singletons typically belong to \mathcal{N} (e.g., \mu(\{x\}) = 0 for Lebesgue measure), and \mathcal{N} is the \sigma-ideal consisting of all measurable sets of measure zero that contains them.^[15]^[16]

Relation to Measurability

In complete measure spaces, every set of measure zero is measurable by definition, as the measure assigns zero to it and includes it in the σ-algebra. The Lebesgue measure on \mathbb{R}^n is complete, meaning that if N is a Lebesgue null set (i.e., has Lebesgue measure zero) and A \subset N, then A is also Lebesgue measurable with measure zero.^[15]^[3] The Lebesgue σ-algebra is the completion of the Borel σ-algebra with respect to Lebesgue measure; this process extends the σ-algebra by adjoining all subsets of Borel null sets, rendering them measurable. In contrast, the Borel σ-algebra is incomplete, so there exist sets with Lebesgue outer measure zero that are not Borel measurable. Such sets qualify as "null" via outer measure but lie outside the Borel structure.^[15] A representative example arises from the middle-thirds Cantor set C \subset [0,1], which is Borel and has Lebesgue measure zero. The power set of C has cardinality $2^{2^{\aleph_0}}, exceeding the cardinality $2^{\aleph_0} of the Borel σ-algebra, so most subsets of C are not Borel; however, all such subsets are Lebesgue measurable with measure zero due to the completeness of Lebesgue measure. Constructions of explicit non-Borel subsets of null sets often invoke the axiom of choice.^[15]

Applications

In Real Analysis

In Lebesgue integration on the real line or \mathbb{R}^n, two measurable functions f and g are considered equal almost everywhere (a.e.) if the set where they differ has Lebesgue measure zero, i.e., they differ on a null set.^[17] This equivalence ensures that if f and g are equal a.e. and one is Lebesgue integrable, then so is the other, and their integrals coincide: \int f \, d\mu = \int g \, d\mu.^[18] Such equivalence classes modulo null sets form the foundation for defining the Lebesgue integral, allowing negligible discrepancies on null sets without affecting the overall integration process.^[19] In the context of L^p spaces for $1 \leq p < \infty, the elements are precisely these equivalence classes of measurable functions that are equal a.e., where the L^p norm is well-defined and finite almost everywhere.^[20] The norm \|f\|_p = \left( \int |f|^p \, d\mu \right)^{1/p} remains unchanged under modifications on null sets, making L^p a Banach space under this identification.^[21] This structure modulo null sets is essential for ensuring that the space captures the essential behavior of functions up to sets of measure zero, facilitating convergence and boundedness arguments in functional analysis.^[22] A pivotal result relying on null sets is Lebesgue's dominated convergence theorem, which addresses pointwise a.e. convergence of sequences of integrable functions.^[23] Specifically, if (f_n) is a sequence of measurable functions converging pointwise a.e. to f, and there exists an integrable function g such that |f_n| \leq g a.e. for all n, then f is integrable and \lim_{n \to \infty} \int f_n \, d\mu = \int f \, d\mu.^[24] The theorem's proof hinges on the negligible impact of null sets, where convergence may fail, to justify interchanging limits and integrals.^[25] The completeness of Lebesgue measure plays a crucial role in real analysis by guaranteeing that every subset of a null set is itself measurable (and null).^[26] This property ensures the measure space is complete, incorporating all such subsets into the \sigma-algebra without introducing non-measurable sets.^[3] It underpins differentiation theorems, such as Lebesgue's density theorem, which states that for any measurable set A \subset \mathbb{R}^n, almost every point in A has density 1 with respect to A, meaning the proportion of A in small balls around the point approaches 1.^[27] Similarly, almost every point outside A has density 0.^[28] This a.e. behavior, protected by completeness, is vital for results in differentiation of integrals and the structure of measurable functions.

In Probability Theory

In probability theory, a null set, or null event, is a measurable set A in a probability space (\Omega, \mathcal{F}, P) such that P(A) = 0. Such events are considered negligible or impossible in a probabilistic sense, though they may be non-empty; their complements have probability 1. This concept arises because probability measures, being special cases of measures, assign zero probability to sets of measure zero, allowing focus on events that occur with positive probability while disregarding sets that do not affect overall outcomes.^[29]^[30] A key application of null sets is the notion of "almost surely" (a.s.), where a property or event holds almost surely if the set of outcomes where it fails is a null set, i.e., has probability zero. For instance, in the Wiener process (standard Brownian motion), the sample paths are continuous almost surely, meaning the probability that a path is discontinuous is zero, even though discontinuous paths exist in the sample space. Similarly, the strong law of large numbers states that the sample average of independent, identically distributed random variables with finite expectation converges almost surely to the true expectation, ignoring convergence failures on a null set. The central limit theorem also holds in probability, with stronger versions implying almost sure properties under certain conditions, emphasizing that null sets enable precise statements about typical behavior.^[31]^[32]^[33] Examples of null events abound in continuous distributions. For a continuous random variable X with a probability density function, the probability of any singleton event P(X = c) = 0 for any constant c, as the cumulative distribution function is absolutely continuous and assigns zero mass to points. Another classic example is the set of rational numbers \mathbb{Q} \cap [0,1] under the uniform distribution on [0,1], which has Lebesgue measure (and thus probability) zero despite being countable and dense, highlighting how null sets can be "small" yet pervasive.^[34]^[30] Null sets play a crucial role in independence and conditional probability, where they do not influence outcomes. Specifically, any event is independent of a null event, as P(A \cap N) = P(N) = 0 = P(A) \cdot 0 for any measurable A, and conditional probabilities are undefined when conditioning on null sets to avoid division by zero, ensuring that negligible events are treated consistently without altering probabilistic relations. This preserves the structure of probability measures under conditioning on positive-probability events.

Advanced Concepts

Measure Completion

In measure theory, the completion of a measure space (X, \Sigma, \mu) addresses the incompleteness that arises when subsets of null sets are not necessarily measurable. The completed \sigma-algebra \overline{\Sigma} consists of all sets of the form E = A \cup N, where A \in \Sigma and N \subseteq B for some null set B \in \Sigma with \mu(B) = 0. This construction incorporates all subsets of null sets into the measurable sets, ensuring that the extended space is complete, meaning every subset of a null set is measurable.^[35] The measure \mu extends uniquely to \overline{\Sigma} by defining \overline{\mu}(E) = \mu(A) for E = A \cup N as above, preserving the original measure on \Sigma. This extension is well-defined because if E = A_1 \cup N_1 = A_2 \cup N_2 with A_1, A_2 \in \Sigma and N_1 \subseteq B_1, N_2 \subseteq B_2 null, then \mu(A_1) = \mu(A_2) due to the symmetric difference being contained in a null set. The completed measure space (X, \overline{\Sigma}, \overline{\mu}) is the smallest complete measure space containing (X, \Sigma, \mu), as any complete extension must include all such sets E.^[35]^[36] A key theorem states that every measure space admits a unique completion (up to the extension of the measure), and \overline{\Sigma} is a \sigma-algebra with \overline{\mu} being a complete measure. For instance, the Lebesgue measure on the Borel \sigma-algebra \mathcal{B}(\mathbb{R}) completes to the full Lebesgue \sigma-algebra \mathcal{L}(\mathbb{R}), which includes all Borel sets and their null subsets, enabling the measurement of a broader class of sets in \mathbb{R}.^[35] For E \in \overline{\Sigma}, the extended measure agrees with the induced outer measure:

\overline{\mu}(E) = \inf \{ \mu(A) \mid A \in \Sigma, \, E \subseteq A \},

which aligns the completion with the outer measure construction and ensures consistency for \sigma-finite measures. This equivalence holds because any covering A \supseteq E must cover both the measurable part and the null addition, with the infimum capturing the minimal measure.^[36] The implications of measure completion are profound in analysis: it systematically adds all subsets of null sets as measurable without altering measures of original sets, facilitating the treatment of non-Borel sets that differ from Borel sets by null sets. This is essential for rigorous handling of limits, integrals, and convergence theorems involving potentially non-measurable negligible sets, ensuring completeness in applications like integration theory.

Haar Null Sets

In a locally compact topological group G equipped with a left (or right) Haar measure \mu, which is a non-zero, positive Borel measure invariant under left (or right) translations, a subset N \subseteq G is defined as Haar null if for every compact subset K \subseteq G and every \varepsilon > 0, there exist countably many elements g_i \in G (for i \in \mathbb{N}) such that N \subseteq \bigcup_{i=1}^\infty g_i K and \mu\left(\bigcup_{i=1}^\infty g_i K\right) < \varepsilon.^[37] This covering condition captures sets that are negligible with respect to the invariant measure, generalizing the notion of measure zero while respecting the group's algebraic structure; equivalently, in locally compact groups, Haar null sets coincide precisely with those of Haar measure zero.^[37] The definition can be adapted to use a fixed Borel probability measure with compact support on compacta to witness the nullness, ensuring invariance under the group action.^[38] The family of Haar null sets forms a \sigma-ideal, meaning it is closed under subsets and countable unions, and contains the empty set.^[37] In non-abelian groups, distinctions arise between left Haar null sets (using left translates g_i K) and right Haar null sets (using right translates K g_i), which may not coincide, unlike in abelian groups where they align; the full (two-sided) Haar null sets require nullness under both left and right (or conjugations) actions.^[37] For instance, in \mathbb{R}^n under addition with Lebesgue measure as the Haar measure, Haar null sets are exactly the Lebesgue null sets, as the covering reduces to standard measure-theoretic coverings.^[37] In contrast, infinite-dimensional separable Banach spaces, which lack a locally compact topology, admit an extension of the Haar null concept via Borel probability measures \mu such that \mu(g B h) = 0 for all g, h \in G and Borel B \supseteq N, revealing strictly more Haar null sets than in finite dimensions, including certain analytic sets without Borel hulls.^[37]^[38] Haar null sets feature prominently in descriptive set theory within Polish groups, where they complement Haar meager sets (the measure-theoretic analog of category smallness) to analyze the Baire category theorem versus invariance principles, often showing that comeager sets can be Haar null or vice versa in non-locally compact settings.^[37] Analogs of the Steinhaus theorem hold: if A \subseteq G is a Borel set that is not Haar null, then the difference set A - A = \{a^{-1} b \mid a, b \in A\} (or sumset in additive notation) contains an open neighborhood of the identity element.^[39] This extends classical results from \mathbb{R}^n to abstract groups, aiding studies of automatic continuity and subgroup structures.^[37] The Haar measure, on which the notion of null sets is based, was introduced by Alfred Haar in his 1933 paper "Der Massbegriff in der Theorie der kontinuierlichen Gruppen," published in the Annals of Mathematics, providing an invariant integration framework on locally compact groups.^[40] John von Neumann established the uniqueness of Haar measure (up to scalar multiples) for separable locally compact groups in 1936, implicitly defining null sets via the measure. The explicit generalization to non-locally compact Polish groups, preserving key properties like \sigma-ideality, was formalized by J. P. R. Christensen in 1972.^[41]

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[38]
[PDF] 5. Outer measures - KSU Math
The correspondence (S,ν) 7− → (¯S, ¯ν) is referred to as the measure completion. ... so taking the infimum yields µ∗(S) ≥ ν(S). D. Proposition 5.5. Let B ...
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[PDF] Haar null and Haar meager sets: a survey and new results - arXiv
We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of ...
[40]
[PDF] Haar null sets without Gδ hulls
Dec 30, 2013 · Definition 1.1 A set X ⊂ G is called Haar ... and also that in locally compact groups a set is Haar null iff it is generalised Haar null.
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[PDF] The Steinhaus property and Haar-null sets
Steinhaus theorem for an arbitrary Polish group. ... A to Haar-null sets, instead of merely left Haar-null ... Therefore, there is no analog of Theorem A for Haar- ...
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On sets of Haar measure zero in abelian polish groups
It is shown that the concept of zero set for the Haar measure can be generalized to abelian Polish groups which are not necessarily locally compact.