# Definition:Measurable Set/Subsets of Real Space

Jump to navigation
Jump to search

## Definition

A subset $S$ of $\R^n$ is said to be **Lebesgue measurable**, frequently just **measurable**, if and only if for every set $A \in \R^n$:

- $m^* A = \map {m^*} {A \cap S} + \map {m^*} {A \cap \map \complement S}$

where:

- $\map \complement S$ is the complement of $S$ in $\R^n$

- $m^*$ 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 $\mathfrak M_{\R^n}$.

## Sources

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.Refactored page now has 3 subpagesIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.1$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**measurable set**