Lebesgue Null Sets: Countable or Not?

Thread starter aaaa202
Start date Oct 4, 2013
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Sets

In summary, a Lebesgue null set is a subset of a larger set with a measure of zero, meaning it contains no measurable points and has no volume. These sets can be either countable or uncountable, depending on their specific properties. They can also contain uncountably many points as long as their measure is still considered to be zero. Lebesgue null sets differ from the traditional notion of "null sets" as they have zero measure in the more general concept of Lebesgue integration. Finally, Lebesgue null sets cannot have positive Lebesgue measure as they, by definition, have a measure of zero.

Oct 4, 2013

aaaa202

Obviously countable sets of R are Lebesgue null sets - but are Lebesgue null sets countable?

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Oct 4, 2013

pwsnafu

Science Advisor

See Cantor set.

Related to Lebesgue Null Sets: Countable or Not?

1. What is a Lebesgue null set?

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.

2. Are Lebesgue null sets countable or uncountable?

Lebesgue null sets can be either countable or uncountable. It depends on the specific set and its properties.

3. Can a Lebesgue null set contain uncountably many points?

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.

4. How are Lebesgue null sets different from the traditional notion of "null sets"?

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.

5. Can a Lebesgue null set have positive Lebesgue measure?

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.