# Equivalence of Definitions of Null Set in Euclidean Space

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of basic complexity.In particular: Restructure into our standard formUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 `{{Refactor}}` from the code. |

## Theorem

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

Let $E \subseteq \R^n$.

Then the following are equivalent:

- $(1):\quad \exists B \in \map \BB {\R^n}: E \subseteq B, \map {\lambda^n} B = 0$
- $(2):\quad$ For every $\epsilon > 0$, there exists a countable cover $\family {J_i}_{i \mathop \in \N}$ of $E$ by open $n$-rectangles, such that:
- $\ds \sum_{i \mathop = 1}^\infty \map {\operatorname {vol} } {J_i} \le \epsilon$

This article, or a section of it, needs explaining.In particular: link to some definition (might not exist) for $\operatorname {vol}$You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.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 `{{Explain}}` from the code. |

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.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 `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 6$: Problem $8$