Definition:Null Set

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

A set $N \in \Sigma$ is called a ($\mu$-)null set if and only if $\map \mu N = 0$.

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Family of Null Sets

The family of $\mu$-null sets, $\set {N \in \Sigma: \map \mu N = 0}$, is denoted $\NN_\mu$.

Signed Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

Let $N \in \Sigma$.

We say that $N$ is a $\mu$-null set if and only if:

for each $A \in \Sigma$ with $A \subseteq N$, we have $\map \mu A = 0$

Definition in $\R^n$

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A set $E \subseteq \R^n$ is called a null set if for any $\epsilon \in \R_{>0}$ there exists a countable collection $J_i := \paren {\openint {\mathbf a_i} {\mathbf b_i} }$, $i \in \N$ of open $n$-rectangles such that:

$\ds E \subseteq \bigcup_{i \mathop = 1}^\infty J_i$

and

$\ds \sum_{i \mathop = 1}^\infty \map {\operatorname{vol} } {J_i} \le \epsilon$.

Here, $\map {\operatorname{vol} } {J_i}$ denotes the volume of the open rectangle $J_i$, which is the product of the lengths of its sides.

Said another way, a null set is a set that can be covered by a countable collection of open $n$-rectangles having total volume as small as we wish.

On Equivalence of Definitions of Null Set in Euclidean Space, it is shown that this definition is compatible with that for general measure spaces.

Examples

Rational Numbers form Null Set under Lebesgue Measure

Let $\lambda$ be $1$-dimensional Lebesgue measure on $\R$.

Let $\Q$ be the set of rational numbers.

Then:

$\map \lambda \Q = 0$

that is, $\Q$ is a $\lambda$-null set.

Also defined as

Many sources use the term null set to mean what is called the empty set on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also known as

Because of the defining equality $\map \mu N = 0$, a $\mu$-null set $N$ is also sometimes called a ($\mu$-)measure zero set.

Also see

• Results about null sets can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $10$