A measurable set is a null set for a measure when . A property holds almost everywhere when its exceptions lie in a null set. With a complete measure, every subset of a null set is measurable and is itself a null set.
A Lebesgue null set has zero Lebesgue measure. Since Lebesgue measure is complete, every subset of one is also measurable and null. The Cantor set is an uncountable example with positive Hausdorff dimension; the rational numbers form a countable, dense example with Hausdorff dimension zero. Being null does not imply being empty or having finitely many points.
A Borel set of Lebesgue measure zero. Every Lebesgue-null set is contained in a Borel null set, by outer regularity; consequently testing all Borel null sets suffices in the definition of a Sierpiński set.
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A null set, also known as an empty set, is a fundamental concept in set theory. It is defined as a set that contains no elements. The null set is typically denoted by the symbol ∅ or by using curly braces, such as {}. Some key points about the null set include: 1. **Unique Set**: There is only one null set, which is unique, meaning that any two null sets are considered to be the same set.