# Definition:Measurable Set/Subsets of Real Space

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## Definition

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 $\mathfrak M_{\R^n}$.

## Sources

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- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.1$