A question on proving countable additivity

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In summary: The proof in the text does not make this clear, but this is the case if the first and last form mentioned in the lemma are used in the proof. This would mean that the entire [a,b] is covered by the constructed intervals \{I_j\}.
  • #1
zzzhhh
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This question comes from the proof of Lemma 9.3 of Bartle's "The Elements of Integration and Lebesgue Measure" in page 97-98. This proof is shown as the image below.
684m80.png


Form (9.1) mentioned in the lemma is: [tex](a,b], (-\infty,b], (a,+\infty), (-\infty,+\infty)[/tex].

My question is: although [tex]I_j[/tex] constructed in P98 is a bit fatter than [tex](a_j,b_j][/tex], I doubt the assertion that the left endpoint a, and in turn the compact interval [a,b], is also covered by [tex]\{I_j\}[/tex], as the proof in the text claimed (I drew a red underline). Is my doubt correct (this means the text is incorrect), or point a can be proved to be covered by [tex]\{I_j\}[/tex] (how)? Thanks!
PS: the establishment of the converse inequality does not need the coverage of the whole [a,b]. A small shrink, say [tex][a+\epsilon,b][/tex], is sufficient to get the inequality.
 
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  • #2
Definitely seems to be an error in the text. If [tex]a_i=a+\frac{1}{n}[/tex] and [tex]\epsilon=\frac{1}{2}[/tex] and [tex]\epsilon_i=\frac{1}{2^{i+1}}[/tex] the [tex]I_j[/tex] never include [tex]a[/tex]

It seems true that you can pick to [tex]\epsilon_i[/tex]'s so that you get a covering... we know that the [tex]a_i[/tex] have to get arbitrarily close to [tex]a[/tex], so you can pick one really close to [tex]a[/tex] to add one of your larger values of [tex]\epsilon_i[/tex] to
 
  • #3
Thank you Office_Shredder!
 
  • #4
zzzhhh said:
This question comes from the proof of Lemma 9.3 of Bartle's "The Elements of Integration and Lebesgue Measure" in page 97-98. This proof is shown as the image below.
684m80.png


Form (9.1) mentioned in the lemma is: [tex](a,b], (-\infty,b], (a,+\infty), (-\infty,+\infty)[/tex].

My question is: although [tex]I_j[/tex] constructed in P98 is a bit fatter than [tex](a_j,b_j][/tex], I doubt the assertion that the left endpoint a, and in turn the compact interval [a,b], is also covered by [tex]\{I_j\}[/tex], as the proof in the text claimed (I drew a red underline). Is my doubt correct (this means the text is incorrect), or point a can be proved to be covered by [tex]\{I_j\}[/tex] (how)? Thanks!
PS: the establishment of the converse inequality does not need the coverage of the whole [a,b]. A small shrink, say [tex][a+\epsilon,b][/tex], is sufficient to get the inequality.

I have not gone through the entire arguement, but the condition 9.2 and the ordering of the a's and b's assumed would require that [tex] a=a_1[/tex] and [tex]b_i=a_{i+1}[/tex].
 

1. What is countable additivity?

Countable additivity is a mathematical property that states that the measure of the union of a countable collection of disjoint sets is equal to the sum of the measures of each individual set. In other words, if A1, A2, A3, ... are disjoint sets, then the measure of their union (A1 ∪ A2 ∪ A3 ∪ ...) is equal to the sum of their individual measures (|A1| + |A2| + |A3| + ...).

2. How is countable additivity related to probability theory?

In probability theory, countable additivity is a fundamental property of a probability measure. It ensures that the total probability of all possible outcomes in a sample space is equal to 1. This property is necessary for the consistency and coherence of probability theory.

3. What are some examples of countable additivity?

Countable additivity can be applied to various mathematical concepts, such as Lebesgue measure, Riemann integral, and probability theory. For example, in probability theory, if the probability of an event A is 0.3 and the probability of an event B is 0.5, then the probability of either A or B occurring is 0.8, which follows the principle of countable additivity.

4. How is countable additivity proven?

The proof of countable additivity depends on the specific mathematical concept or theory in which it is being applied. Generally, it involves breaking down a larger set or measure into smaller, disjoint sets and then showing that their measures can be added together to equal the measure of the original set. This can be done using mathematical induction, set theory, or other methods depending on the context.

5. What are some implications of countable additivity?

Countable additivity has many important implications in mathematics, particularly in areas such as measure theory and probability theory. It ensures that certain mathematical constructions are well-defined and consistent, and it allows for the extension of mathematical concepts to infinite sets or measures. It also has practical applications in fields such as economics, physics, and engineering, where it is used to model and analyze real-world systems.

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