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If N is of null Lebesgue measure. Can we find a Borel set B of null measure such that N is entirely contained in B?
Lebesgue and Borel sigma algebras are both types of mathematical structures used to measure sets. The main difference between them is the types of sets they can measure. Lebesgue sigma algebra can measure more sets than Borel sigma algebra, as it includes all the sets that Borel sigma algebra can measure and some additional sets as well.
Borel sigma algebra is a subset of Lebesgue sigma algebra, meaning that all sets measurable by Borel sigma algebra are also measurable by Lebesgue sigma algebra. In other words, Borel sigma algebra is a special case of Lebesgue sigma algebra.
Lebesgue and Borel sigma algebras are fundamental concepts in measure theory, which is a branch of mathematics that deals with the concept of measuring sets. These sigma algebras provide a way to measure various types of sets, which is crucial in many areas of mathematics, such as probability theory and analysis.
The Lebesgue measure is a measure defined on the Lebesgue sigma algebra, while the Borel measure is a measure defined on the Borel sigma algebra. Both measures assign a non-negative value to each set in their respective sigma algebra, with the Lebesgue measure being more comprehensive.
Yes, the Cantor set is an example of a set that is measurable by Lebesgue sigma algebra but not by Borel sigma algebra. The Cantor set is a fractal set with a complicated structure, and it is not a Borel set, meaning it cannot be measured by Borel sigma algebra. However, it is measurable by Lebesgue sigma algebra, as it is a subset of the real numbers and can be defined using intervals, which are measurable by Lebesgue sigma algebra.