Null Sets and Almost Everywhere — Mathematics Knowledge System

Overview

A null set (or measure-zero set) is a measurable set whose measure is zero. The phrase 'almost everywhere' (abbreviated a.e.) means 'except possibly on a null set'. These concepts are ubiquitous in analysis and probability: two functions that agree a.e. are identified in L^p spaces, convergence theorems hold a.e., and properties that fail only on null sets are treated as universally true for most analytic purposes.

Intuition

A null set is so 'small' in the measure-theoretic sense that it contributes nothing to any integral. The Cantor set is a striking example: it is uncountable (as large as R in cardinality) yet has Lebesgue measure zero. 'Almost everywhere' is the measure-theoretic analogue of 'for all practical purposes': if a property holds except on a null set, integrals and limits are unaffected.

Formal Definition

Definition

Let (X, M, mu) be a measure space. A set N in M is a null set if mu(N) = 0. A property P(x) holds almost everywhere (mu-a.e.) if the set {x : P(x) fails} is contained in a null set. When the measure is understood, we write 'a.e.' without qualification.

N \text{ is a null set} \iff N \in \mathcal{M} \text{ and } \mu(N) = 0

Null set definition

P(x) \text{ holds } \mu\text{-a.e.} \iff \mu(\{x \in X : P(x) \text{ fails}\}) = 0

Almost everywhere definition

Notation

Notation	Meaning
$\text{a.e.}$	Almost everywhere (except on a null set)
$f = g \;\mu\text{-a.e.}$	f and g agree except on a set of measure zero
$\mathcal{N}(\mu)$	The collection of all mu-null sets

Properties

Countable union of null sets is null

\text{If } \mu(N_k) = 0 \text{ for all } k \in \mathbb{N}, \text{ then } \mu\!\left(\bigcup_{k=1}^{\infty} N_k\right) = 0.

Subsets of null sets are null (completeness)

\text{If } \mu(N) = 0 \text{ and } E \subseteq N, \text{ then in a complete measure space } \mu(E) = 0.

Condition: Requires the measure space to be complete

The Cantor set has measure zero

\lambda(C) = 0, \text{ where } C \text{ is the standard Cantor set and } \lambda \text{ is Lebesgue measure.}

Countable sets are null

\text{If } E \subseteq \mathbb{R} \text{ is countable, then } \lambda(E) = 0.

Worked Examples

Construct C by removing open middle thirds iteratively. At stage n, we remove 2^{n-1} intervals each of length 1/3^n.
$\text{Total length removed} = \sum_{n=1}^{\infty} \frac{2^{n-1}}{3^n} = \frac{1}{3} \cdot \frac{1}{1 - 2/3} = 1$
Since [0,1] has measure 1 and we removed measure 1 in total, the remaining set C has measure 0.
$\lambda(C) = \lambda([0,1]) - 1 = 1 - 1 = 0$
C is uncountable because it is in bijection with {0,1}^N via ternary expansions using digits 0 and 2.

Answer: The Cantor set has Lebesgue measure zero (all measure was removed in the construction) but is uncountable.

Practice Problems

Difficulty 5/10

Prove that every countable subset of R has Lebesgue measure zero.

Difficulty 6/10

Give an example of a function f: [0,1] -> R that is discontinuous on a dense set yet Riemann integrable, and explain the role of null sets.

Difficulty 5/10

Which statement about null sets is FALSE?

A countable union of null sets is nullAn uncountable set can have measure zeroA null set must be countableSubsets of null sets are null in a complete measure space

Common Mistakes

Common Mistake

Null sets must be countable.

The Cantor set is a canonical counterexample: uncountable yet Lebesgue measure zero. Measure zero reflects smallness in the measure-theoretic sense, not in cardinality.

Common Mistake

'Almost everywhere' means 'on a set of full measure', i.e., exceptions have measure zero but the exceptional set might not be measurable.

In the standard definition, P holds a.e. iff {x : P(x) fails} is contained in a measurable null set. In complete measure spaces every subset of a null set is measurable and null, so the distinction dissolves.

Common Mistake

If f = g a.e. then f and g are equal as functions.

They may differ on a null set. In L^p spaces, f and g are identified as the same equivalence class, but as pointwise functions they can differ.

Quiz

Historical Background

Henri Lebesgue introduced null sets in his 1902 thesis as part of his construction of the Lebesgue integral. The phrase 'almost everywhere' crystallised in the early 20th century as analysts recognised that measure-zero exceptions are irrelevant to integration.

1902
Lebesgue defines measure-zero sets and integration ignoring them
Henri Lebesgue
1910s
'Almost everywhere' becomes standard terminology in analysis

Summary

A null set is a measurable set of measure zero; almost everywhere (a.e.) means except on a null set.
Countable unions of null sets are null; countable sets are null under Lebesgue measure.
The Cantor set is the archetypal uncountable null set, showing measure-zero does not imply countability.
Functions agreeing a.e. are identified in L^p spaces; most convergence theorems hold a.e. rather than everywhere.
Complete measure spaces include all subsets of null sets as measurable null sets, eliminating pathological exceptions.

References

BookRudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill.
BookRoyden, H. L., & Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson.
WebsiteWikipedia — Null set