# Null set

A null set is a set of measure zero: in the real line, a Lebesgue measurable set that can be covered by countably many open intervals whose total length is arbitrarily small.<sup>[1](https://maths.free/measure-theory/null-set/)</sup> Null sets are the sets that integration and probability theory agree to ignore, and the phrase "almost everywhere" means exactly "everywhere except on a null set".<sup>[2](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/02%3A_Probability_Spaces/2.07%3A_Measure_Spaces)</sup>

| Key fact | Detail |
|---|---|
| Definition | A Lebesgue measurable set of real numbers with measure zero; equivalently, coverable by countably many open intervals of total length at most \( \varepsilon \) for every \( \varepsilon > 0 \)<sup>[1](https://maths.free/measure-theory/null-set/)</sup> |
| Countable sets | Every finite or countably infinite subset of \( \mathbb{R} \) is null<sup>[1](https://maths.free/measure-theory/null-set/)</sup> |
| Closure properties | Countable unions of null sets are null, and in a complete measure space every subset of a null set is null; the null sets form a sigma-ideal<sup>[3](https://alphabetamath.cc/item/prop-null-sets-form-a-sigma-ideal-in-a-complete-space)</sup> |
| Uncountable null sets | The Cantor middle-thirds set is uncountable yet has Lebesgue measure zero<sup>[4](https://www.math.hkust.edu.hk/~majhu/Math3043/Lebesgue_Measure.pdf)</sup> |
| Limits of smallness | No interval of positive length is null, and the Smith-Volterra-Cantor set is nowhere dense without being null<sup>[5](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)</sup> |
| Role in integration | A bounded function on \([a,b]\) is Riemann integrable exactly when its discontinuity set is null (Lebesgue's criterion)<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup> |
| Measure dependence | Nullness is relative to the chosen measure; the rationals are null for Lebesgue measure but can be non-null for another measure, and vice versa<sup>[7](https://link.springer.com/chapter/10.1007/978-3-030-20103-6_3)</sup> |

## Definition

Fix a measure space \((\Omega, \mathcal{A}, \mu)\). A measurable set \(A \in \mathcal{A}\) is a null set when \(\mu(A) = 0\).<sup>[2](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/02%3A_Probability_Spaces/2.07%3A_Measure_Spaces)</sup> On the real line with [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure), this says that for every \(\varepsilon > 0\) there is a countable family of open intervals \((I_s)\) covering the set with \(\sum_s \ell(I_s) \le \varepsilon\), where \(\ell(I)\) is the interval's length.<sup>[1](https://maths.free/measure-theory/null-set/)</sup> The definition requires a sequence of such covers whose total lengths tend to zero.<sup>[1](https://maths.free/measure-theory/null-set/)</sup>

The covering formulation goes back to the transition from finite to countable covers. Jordan content, introduced by Peano in 1887 and Jordan in 1892, covered sets by finitely many intervals; Borel in 1898 and Lebesgue in 1902 allowed countably many, which is what makes countable sets like the rationals measure zero and produces the Lebesgue integral.<sup>[8](https://www.slideshare.net/slideshow/measurenotesch2pdf/252066450)</sup> Lebesgue presented the theory in his 1902 doctoral thesis *Intégrale, Longueur, Aire*.<sup>[9](https://repositorio.unesp.br/entities/publication/6f8d7bd4-f687-40c7-b0bf-267fdb9a61ba)</sup>

## The sigma-ideal of null sets

Null sets behave like an ideal of negligible objects. Countable subadditivity, \(\mu\left(\bigcup_i A_i\right) \le \sum_i \mu(A_i)\), gives directly that a countable union of null sets is null: if each \(\mu(A_i) = 0\) the sum on the right is zero.<sup>[10](https://math.stackexchange.com/questions/132267/countable-union-of-measure-0-sets-has-measure-0)</sup> Monotonicity gives that any measurable subset of a null set is null.<sup>[4](https://www.math.hkust.edu.hk/~majhu/Math3043/Lebesgue_Measure.pdf)</sup>

**Completeness matters.** In a general measure space, a subset of a null set need not itself be measurable, so "subset of a null set" and "null set" can differ. In a [complete measure space](https://www.edgepedia.org/complete-measure-space), such as Lebesgue measure on its full sigma-algebra, every subset of a null set is measurable and null, and the null sets together with countable unions form a sigma-ideal.<sup>[3](https://alphabetamath.cc/item/prop-null-sets-form-a-sigma-ideal-in-a-complete-space)</sup> This is why Lebesgue measure is completed by adjoining all subsets of Borel null sets.<sup>[3](https://alphabetamath.cc/item/prop-null-sets-form-a-sigma-ideal-in-a-complete-space)</sup>

## Examples and counterexamples

**Countable sets.** A single point \(\{a\}\) is contained in \((a - \varepsilon/2,\ a + \varepsilon/2)\), an interval of length \(\varepsilon\), so its outer measure is at most \(\varepsilon\) for every \(\varepsilon\), hence zero. A countable set is a countable union of points, so it is null by subadditivity.<sup>[4](https://www.math.hkust.edu.hk/~majhu/Math3043/Lebesgue_Measure.pdf)</sup> In particular the rationals \(\mathbb{Q}\) are a null set even though they are dense in \(\mathbb{R}\).<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup>

**The Cantor set.** The middle-thirds [Cantor set](https://www.edgepedia.org/cantor-set) \(C\) is uncountable, yet null. At stage \(n\) of its construction it is contained in \(2^{n}\) closed intervals each of length \(3^{-n}\); enlarging each slightly gives an open cover of total length \((2/3)^{n} + \varepsilon\), which can be made as small as desired.<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup> So nullness does not imply countability, and countability is strictly stronger than measure zero.<sup>[4](https://www.math.hkust.edu.hk/~majhu/Math3043/Lebesgue_Measure.pdf)</sup>

**What cannot be null.** No interval of positive length is null, since its measure equals its length.<sup>[5](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)</sup> The Smith-Volterra-Cantor set, a "fat" Cantor set, is compact, perfect, and nowhere dense but has positive measure, so it is not null.<sup>[5](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)</sup>

**Dependence on the measure.** Nullness is a property of a set together with a measure, not of the set alone. Under a suitable Stieltjes measure the rationals form a null set even though they are dense; under Dirac measure \(\delta_a\), every set not containing \(a\) is null. The same function can be almost-everywhere continuous for one measure and not for another, so the measure must be chosen with the application in mind.<sup>[7](https://link.springer.com/chapter/10.1007/978-3-030-20103-6_3)</sup>

## Null sets and Baire category

Measure zero is one of two standard senses in which a subset of the real line can be small. The other is topological: a set is nowhere dense if its closure has empty interior, and meager (of first category) if it is a countable union of nowhere dense sets. Every countable set is both meager and null.<sup>[11](https://cs.pwr.edu.pl/cichon/2025_26_a/TOPMEASURE/Books/MeasureAndCategory.pdf)</sup>

The two notions are independent. The Cantor set is small in both senses: uncountable, meager, and of measure zero.<sup>[5](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)</sup> But neither implication holds in general. The Smith-Volterra-Cantor set is nowhere dense, hence meager, without being null, so meager does not imply null.<sup>[5](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)</sup> This is why analysts keep both vocabularies: a property can hold outside a null set while failing on a residual set, or the reverse.

## Role in integration and probability

The practical purpose of null sets is to make "ignore this set" rigorous. A statement about points of a measure space holds almost everywhere if it holds except on a null set; under a probability measure the same phrase becomes almost surely, or with probability 1.<sup>[2](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/02%3A_Probability_Spaces/2.07%3A_Measure_Spaces)</sup> Because countable unions of null sets are null, a countable list of almost-everywhere statements still holds almost everywhere.<sup>[10](https://math.stackexchange.com/questions/132267/countable-union-of-measure-0-sets-has-measure-0)</sup>

**Lebesgue's criterion.** The classical application to Riemann integration is Lebesgue's theorem: a bounded function \(f : [a,b] \to \mathbb{R}\) is Riemann integrable if and only if it is bounded and its set of discontinuities has Lebesgue measure zero.<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup> This explains why some highly discontinuous functions are still Riemann integrable. The characteristic function \(\chi_C\) of the Cantor set is discontinuous at every point of \(C\), an uncountable set, yet \(C\) is null, so \(\chi_C\) is Riemann integrable on \([0,1]\) with integral zero.<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup> By contrast the Dirichlet function, discontinuous on all of \([0,1]\), is not Riemann integrable.<sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup>

Changing a bounded function on a null set can destroy its Riemann integrability; for example, changing the zero function to 1 at rational points produces the non-Riemann-integrable Dirichlet function, which shows that null sets do matter in Riemann integration.<sup>[12](https://a-warded.org/open-book/university-real-analysis/lesson-42)</sup><sup> • </sup><sup>[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)</sup>

## Variants and refinements

Several strengthenings of measure zero appear in the literature. A set of real numbers is full null when for every \(\varepsilon > 0\) it is contained in an open set \(G\) with \(\lambda(G) < \varepsilon\) in a uniform sense; every set of Lebesgue measure zero is full null.<sup>[13](https://classicalrealanalysis.info/documents/toti-June2012.pdf)</sup> In the Sobolev-scale setting, a set of Lebesgue measure zero is \(s\)-null for every \(s \ge 0\), because \(H^{s}(\mathbb{R}^{n})\) embeds continuously in \(L^{2}(\mathbb{R}^{n})\) for \(s \ge 0\).<sup>[14](https://www.reading.ac.uk/maths-and-stats/-/media/project/uor-main/schools-departments/maths/documents/preprint-13-17-hewett-mar15.pdf?hash=1EB80712ED9633F6D63B0EA886FF6FCD&la=en)</sup>

Nullness also has geometric characterizations. For a [Borel set](https://www.edgepedia.org/borel-set) \(A \subset \mathbb{R}^{n}\) and an \(m\)-dimensional affine subspace \(W(x)\) through \(x\) varying in a Lipschitz way, \(A\) is Lebesgue null if and only if \(A \cap W(x)\) has \(m\)-dimensional [Hausdorff measure](https://www.edgepedia.org/hausdorff-measure) zero for almost every \(x\).<sup>[15](https://arxiv.org/abs/1904.12276)</sup> Recent work extends nullness conclusions to new families of sets: the boundary of every multigeometric Cantorval is a Lebesgue null set.<sup>[16](https://arxiv.org/abs/2510.21878)</sup>

## References

1. [Null set, step-by-step with animation, maths.free](https://maths.free/measure-theory/null-set/)
2. [Measure Spaces, Statistics LibreTexts (Siegrist)](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/02%3A_Probability_Spaces/2.07%3A_Measure_Spaces)
3. [Null sets form a sigma-ideal in a complete space, Alphabeta Math](https://alphabetamath.cc/item/prop-null-sets-form-a-sigma-ideal-in-a-complete-space)
4. [Lebesgue Measure lecture notes, HKUST Math3043](https://www.math.hkust.edu.hk/~majhu/Math3043/Lebesgue_Measure.pdf)
5. [The Cantor Set, Baire Category, and Measure Zero in R, Alphabeta Math](https://alphabetamath.cc/real-analysis/cantor-set-baire-and-measure-zero)
6. [Introduction to Analysis, Chapter 11: The Riemann Integral, UC Davis](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf)
7. [Measures and Probabilities, Springer](https://link.springer.com/chapter/10.1007/978-3-030-20103-6_3)
8. [Measure Theory notes, Chapter 2, SlideShare](https://www.slideshare.net/slideshow/measurenotesch2pdf/252066450)
9. [Integrale, Longueur, Aire de Henri Lebesgue, UNESP repository](https://repositorio.unesp.br/entities/publication/6f8d7bd4-f687-40c7-b0bf-267fdb9a61ba)
10. [Countable union of measure 0 sets has measure 0, Mathematics Stack Exchange](https://math.stackexchange.com/questions/132267/countable-union-of-measure-0-sets-has-measure-0)
11. [Measure and Category, Graduate Texts in Mathematics 2](https://cs.pwr.edu.pl/cichon/2025_26_a/TOPMEASURE/Books/MeasureAndCategory.pdf)
12. [Lebesgue's Criterion: Sets of Measure Zero and Integrability, A-Warded](https://a-warded.org/open-book/university-real-analysis/lesson-42)
13. [Theory of the Integral, ClassicalRealAnalysis](https://classicalrealanalysis.info/documents/toti-June2012.pdf)
14. [s-null sets and Sobolev spaces, University of Reading preprint](https://www.reading.ac.uk/maths-and-stats/-/media/project/uor-main/schools-departments/maths/documents/preprint-13-17-hewett-mar15.pdf?hash=1EB80712ED9633F6D63B0EA886FF6FCD&la=en)
15. [On Lebesgue null sets, Thierry De Pauw, arXiv:1904.12276](https://arxiv.org/abs/1904.12276)
16. [The Lebesgue measure of boundaries of multigeometric Cantorvals, arXiv:2510.21878](https://arxiv.org/abs/2510.21878)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Measure theory*

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