Definition:Measurable Set
Definition
Let $\struct {X, \Sigma}$ be a measurable space.
A subset $S \subseteq X$ is said to be ($\Sigma$-)measurable if and only if $S \in \Sigma$.
Measurable Set of an Arbitrary Outer Measure
Let $\mu^*$ be an outer measure on $X$.
A subset $S \subseteq X$ is called $\mu^*$-measurable if and only if it satisfies the Carathéodory condition:
- $\map {\mu^*} A = \map {\mu^*} {A \cap S} + \map {\mu^*} {A \setminus S}$
for every $A \subseteq X$.
Measurable Subset of the Reals
A subset $S$ of the real numbers $\R$ is said to be Lebesgue measurable, or frequently just measurable, if and only if for every set $A \subseteq \R$:
- $\map {\lambda^*} A = \map {\lambda^*} {A \cap S} + \map {\lambda^*} {A \setminus S}$
where $\lambda^*$ is the Lebesgue outer measure.
The set of all measurable sets of $\R$ is frequently denoted $\mathfrak M_\R$ or just $\mathfrak M$.
Measurable Subset of $\R^n$
A subset $S$ of $\R^n$ is said to be Lebesgue measurable, frequently just measurable, if and only if for every set $A \subseteq \R^n$:
- $m^* A = \map {m^*} {A \cap S} + \map {m^*} {A \setminus S}$
where:
- $A \setminus S$ denotes the set difference between $A$ and $S$
- $m^*$ is defined as:
- $\ds \map {m^*} S = \inf_{\set {I_k}: S \mathop \subseteq \cup I_k} \sum \map v {I_k}$
where:
- $\set {I_k}$ are a sequence of sets satisfying:
- $I_k = \closedint {a_1} {b_1} \times \dots \times \closedint {a_k} {b_k}$
- $\map v {I_n}$ is the volume $\ds \prod_{i \mathop = 1}^n \size {b_i - a_i}$
The set of all measurable sets of $\R^n$ is frequently denoted $\mathfrak M_{\R^n}$.
Also presented as
Some sources present the Carathéodory condition as:
- $\map {\mu^*} A = \map {\mu^*} {A \cap S} + \map {\mu^*} {A \cap \map \complement S}$
rather than using the $A \setminus S$ form.
While $A \cap \map \complement S$ is more unwieldy and cumbersome than $A \setminus S$, it does present the Carathéodory condition in a neatly symmetrical form.
Also see
- Vitali Set Existence Theorem: from the axiom of choice, it is demonstrated that there exist non-measurable subsets of $\R$.
- Results about measurable sets can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): measure
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): measure
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): measurable set
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