# Convergence in measure

Convergence in measure is a mode of convergence for sequences of measurable functions: a sequence \( f_n \) converges in measure to a function \( f \) if, for every \( \eta > 0 \), the measure of the set where \( |f_n(x) - f(x)| > \eta \) tends to zero as \( n \to \infty \).<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> F. Riesz introduced the notion in 1909, and other authors called it asymptotic convergence; its importance grew after Kolmogorov gave probability theory its measure-theoretic interpretation, under which convergence in probability, as in Bernoulli's law of large numbers, is the same as convergence in measure.<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( f_n \to f \) in measure when \( \lim_{n \to \infty} \mu\{x : |f_n(x) - f(x)| > \varepsilon\} = 0 \) for every \( \varepsilon > 0 \)<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup><sup> • </sup><sup>[3](https://do.csc.knu.ua/marynych/wp-content/uploads/sites/2/2023/01/lecture2.pdf)</sup> |
| Origin | Introduced by F. Riesz in 1909, later called asymptotic convergence by other authors<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup> |
| Probability meaning | Under the measure-theoretic interpretation of probability, convergence in probability is the same as convergence in measure<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup> |
| From a.e. convergence | If \( m(E) < \infty \) and \( f_n \to f \) pointwise a.e. on \( E \), then \( f_n \to f \) in measure on \( E \)<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> |
| Riesz theorem | If \( f_n \to f \) in measure, some subsequence \( \{f_{n_k}\} \) converges to \( f \) almost everywhere<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> |
| Converse fails | The dyadic-interval example \( f_n = \chi_{I_{j,k}} \) on \([0,1]\) converges in measure to 0, yet \( \{f_n(x)\} \) converges for no point \( x \in [0,1] \)<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> |
| Place among modes | Almost-sure convergence and \( L^2 \) convergence each imply convergence in probability, which implies convergence in distribution; almost-sure and \( L^2 \) convergence do not imply each other<sup>[4](https://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf)</sup> |

## Definition and basic properties

Let \( \{f_n\} \) be a sequence of measurable functions, finite almost everywhere, on a set \( E \), and let \( f \) be measurable and finite almost everywhere on \( E \). Then \( f_n \to f \) in measure on \( E \) provided that for every \( \eta > 0 \),

\[ \lim_{n \to \infty} m(\{x \in E : |f_n(x) - f(x)| > \eta\}) = 0. \]

The assumptions that the functions be finite almost everywhere make the difference \( |f_n(x) - f(x)| \) meaningful almost everywhere, and measurability guarantees that the sets in the definition are measurable.<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> The same definition is stated for general measures \( \mu \): \( f_n \) converges in \( \mu \)-measure to \( f \) when \( \mu\{x : |f_n(x) - f(x)| > \varepsilon\} \to 0 \) for every \( \varepsilon > 0 \).<sup>[3](https://do.csc.knu.ua/marynych/wp-content/uploads/sites/2/2023/01/lecture2.pdf)</sup>

**Pointwise convergence implies it on finite-measure sets.** If \( m(E) < \infty \) and \( f_n \to f \) pointwise almost everywhere on \( E \) with \( f \) finite almost everywhere, then \( f_n \to f \) in measure on \( E \).<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> The same theorem is stated for general measures: almost-everywhere convergence to \( f \) implies convergence in \( \mu \)-measure.<sup>[3](https://do.csc.knu.ua/marynych/wp-content/uploads/sites/2/2023/01/lecture2.pdf)</sup> The finiteness hypothesis matters, and the proposition raises the question of what happens when \( m(E) = \infty \).<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup>

**The converse fails.** Convergence in measure does not imply pointwise convergence. Define the dyadic subintervals of \([0,1]\) by \( I_{j,k} = [2^{-j}k, 2^{-j}(k+1)] \) for integers \( j \ge 0 \) and \( k = 0, 1, \ldots, 2^j - 1 \), and map each pair \( (j,k) \) injectively to an index \( n = 2^j + k \), enumerating all natural numbers. If \( f_n = \chi_{I_{j,k}} \) for the corresponding pair, then \( f_n \to 0 \) in measure, but for every \( x \in [0,1] \) the sequence \( \{f_n(x)\} \) does not converge.<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup>

## The Riesz theorem and completeness

**Subsequence extraction.** The Riesz theorem states that if \( f_n \to f \) in measure on \( E \), then there exists a subsequence \( \{f_{n_k}\} \) such that \( f_{n_k} \to f \) almost everywhere on \( E \).<sup>[1](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)</sup> This is the standard route for recovering pointwise information from measure convergence: the mode itself controls only the size of the exceptional sets, not the behavior at any fixed point.

**Cauchy in measure.** This form of convergence enjoys the following completeness property: whenever a sequence \( \{f_n\} \) is Cauchy in measure, one can find a measurable function \( f \) together with a subsequence \( g_j = f_{n_j} \) of \( \{f_n\} \) for which \( g_j \) converges almost everywhere to \( f \) as \( j \to \infty \).<sup>[5](https://mathweb.ucsd.edu/~bdriver/280_09-10/Lecture_Notes/Old%20Versions/W2010p2a.pdf)</sup>

## Relation to other modes of convergence

For random variables, Billingsley lists four major modes of convergence: convergence in distribution, convergence in probability, convergence with probability 1 (almost-sure convergence), and convergence in \( L^2 \). Neither of almost-sure and \( L^2 \) convergence implies the other, but each implies convergence in probability, and convergence in probability implies convergence in distribution.<sup>[4](https://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf)</sup> [Convergence in probability](https://www.edgepedia.org/convergence-in-probability) is the probabilistic name for convergence in measure.<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup>

The hierarchy appears in the laws of large numbers: in connection with the law of large numbers, convergence in probability is called the "weak" law and almost-sure convergence the "strong" law; the two have the same hypotheses and the same limit and differ only in the mode of convergence.<sup>[4](https://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf)</sup><sup> • </sup><sup>[6](https://youyuanwu.github.io/misc-docs/math/11-probability/06-convergence-modes.html)</sup> Instances of convergence in distribution often cannot be strengthened as they stand to convergence in probability.<sup>[4](https://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf)</sup>

A consequence used in probability theory is that if a sequence \( \{f_n\} \) of measurable functions converges in measure to \( f \), then the distribution functions \( F_n \) of \( f_n \) converge to the distribution function \( F \) of \( f \) at each continuity point of \( F \).<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup>

## Related theorems

Egoroff's theorem says that pointwise convergence is nearly uniform, and the dominated convergence theorem holds for convergence in measure; the MIT lecture notes covering these results also list the implications almost-everywhere convergence implies convergence in measure, and convergence in measure implies some subsequence converges almost everywhere.<sup>[7](https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/resources/18125_lec13/)</sup> Hartman and Marczewski, studying convergence in measure, gave a condition stronger than the distribution-function theorem above that is not only necessary but also sufficient for the convergence of distribution functions.<sup>[2](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)</sup>

## References

1. [Convergence in measure, lecture notes, Math 776, George Mason University (J. Walnut)](https://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec11_notes.pdf)
2. [S. Hartman and E. Marczewski, "On the convergence in measure," Wrocław](https://acta.bibl.u-szeged.hu/13596/1/math_012_pars_a_125-131.pdf)
3. [Advanced probability theory, Lecture 2, Taras Shevchenko National University of Kyiv](https://do.csc.knu.ua/marynych/wp-content/uploads/sites/2/2023/01/lecture2.pdf)
4. [Patrick Billingsley, Convergence of Probability Measures, 2nd ed., Chapter 5, Section 20](https://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf)
5. [Driver, Lecture Notes 12.2: Almost Everywhere and Measure Convergence, UCSD](https://mathweb.ucsd.edu/~bdriver/280_09-10/Lecture_Notes/Old%20Versions/W2010p2a.pdf)
6. [Youyuan Wu, Modes of Convergence](https://youyuanwu.github.io/misc-docs/math/11-probability/06-convergence-modes.html)
7. [MIT OpenCourseWare 18.125 Measure and Integration, Lecture 13 notes](https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/resources/18125_lec13/)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Convergence of random variables › Relationships among modes of convergence*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license