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Obviously countable sets of R are Lebesgue null sets - but are Lebesgue null sets countable?
A Lebesgue null set is a subset of a larger set with the property that it has a measure of zero. In other words, it contains no measurable points and has no volume.
Lebesgue null sets can be either countable or uncountable. It depends on the specific set and its properties.
Yes, a Lebesgue null set can contain uncountably many points as long as the set has a measure of zero. This means that even if the set contains infinitely many points, its measure is still considered to be zero.
The traditional notion of null sets refers to sets with zero measure in the sense of Riemann integration. Lebesgue null sets, on the other hand, have zero measure in the sense of Lebesgue integration, which is a more general and flexible concept of measuring sets.
No, a Lebesgue null set cannot have positive Lebesgue measure. This is because by definition, a Lebesgue null set has a measure of zero, meaning it contains no measurable points and has no volume.