Measure theory question: Countable sub-additivity

In summary, the property of sub-additivity for sets ##E## and ##E_j## states that if ##E=\bigcup_{j=0}^{\infty}E_j##, then ##m^*(E) \leq \sum_{j=0}^{\infty}m^*(E_j)##. However, the reverse inequality is not true in general. This is because ##m^* \left (\bigcup_{j=0}^{\infty}E_j \right ) \neq \sum_{j=0}^{\infty} m^*(E_j)##, which can be seen by the fact that ##\bigcup_{j=0}
  • #1
Cascabel
2
0
I have a question on sub-additivity. For sets ##E## and ##E_j##, the property states that if

##E=\bigcup_{j=0}^{\infty}E_j##

then

##m^*(E) \leq \sum_{j=0}^{\infty}m^*(E_j)##, where ##m^*(x)## is the external measure of ##x##.

Since ##E\subset \bigcup_{j=0}^{\infty}E_j##, by set equality, the property seems to follow from monotonicity.

However, it is also true that, ##\bigcup_{j=0}^{\infty} E_j \subset E##, which seems to imply the reverse inequality, ##\sum_{j=0}^{\infty} m^*(E_j)\leq m^*(E)##, which is not true.

What's wrong?
 
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  • #2
Sorry, figured it out.

Sorry, figured it out, ##m^* \left (\bigcup_{j=0}^{\infty}E_j \right ) \neq \sum_{j=0}^{\infty} m^*(E_j)##.
 

FAQ: Measure theory question: Countable sub-additivity

What is countable sub-additivity in measure theory?

Countable sub-additivity is a property of a measure function in measure theory. It states that the measure of the union of countably many disjoint sets is equal to the sum of the individual measures of each set. In other words, the measure of a countable union is less than or equal to the sum of the individual measures.

Why is countable sub-additivity important?

Countable sub-additivity is an essential property in measure theory because it allows us to extend the measure function from finite sets to countable sets. This is crucial in many applications, such as probability theory and integration, where we often need to measure countable collections of sets.

Can you give an example of a measure that is not countable sub-additive?

Yes, the counting measure on the set of real numbers is an example of a measure that is not countable sub-additive. This measure assigns the value of 1 to each singleton set, so the measure of the union of countably many disjoint singleton sets would be infinite, violating the property of countable sub-additivity.

What is the difference between countable sub-additivity and countable additivity?

Countable sub-additivity and countable additivity are two related but distinct properties in measure theory. Countable sub-additivity states that the measure of a countable union is less than or equal to the sum of the individual measures, while countable additivity states that the measure of a countable union is equal to the sum of the individual measures. In other words, countable additivity is a stronger condition than countable sub-additivity.

Are all measures countable sub-additive?

No, not all measures are countable sub-additive. For example, the Dirac delta measure assigns a value of 1 to a single point and 0 to all other points, so it is not countable sub-additive. However, many commonly used measures, such as the Lebesgue measure, are countable sub-additive and satisfy this property.

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