Sigma Algebras .... Axler Page 26 ....

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In summary, the conversation revolves around the definition of a $\sigma$-algebra in the context of measure theory. The set of all subsets of $\mathbb{R}$ is mentioned as an example of a $\sigma$-algebra that is not useful in this context. The speaker asks for help in understanding why this is the case, to which the respondent replies that while the set of all subsets of $\mathbb{R}$ does meet the definition of a $\sigma$-algebra, it is not useful in measure theory. The conversation ends with the speaker requesting further explanation on why this is the case.
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I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need help in order to fully understand the implications of Axler's definition of a $\sigma$-algebra ... ...

The relevant text reads as follows:
Axler - Sigma Algebres ... Page 26 .png
Now in the above text Axler implies that the set of all subsets of $\mathbb{R}$ is not a $\sigma$-algebra ... ...

... BUT ... which of the three bullet points of the definition of a $\sigma$-algebra is violated by the set of all subsets of $\mathbb{R}$ ... and how/why is it violated ...
Help will be much appreciated ...

Peter
 
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The set of all subsets of $\Bbb{R}$ is a $\sigma$-algebra (but it is not a useful $\sigma$-algebra in the context of measure theory).
 
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Opalg said:
The set of all subsets of $\Bbb{R}$ is a $\sigma$-algebra (but it is not a useful $\sigma$-algebra in the context of measure theory).
Thanks for the help Opalg ... but can you help further ...

... can you explain what you mean by your statement that the set of all subsets of $\Bbb{R}$ is not a useful $\sigma$-algebra in the context of measure theory ...

Peter
 

Related to Sigma Algebras .... Axler Page 26 ....

1. What is a Sigma Algebra?

A Sigma Algebra is a collection of subsets of a given set that satisfies certain properties, such as being closed under countable unions and complements. It is an important concept in measure theory and probability theory.

2. Why are Sigma Algebras important?

Sigma Algebras are important because they allow us to define and work with measures, which are used to assign a numerical value to a set. This is crucial in many areas of mathematics, including probability, statistics, and analysis.

3. How do you construct a Sigma Algebra?

A Sigma Algebra can be constructed by starting with a collection of subsets of a set and then applying closure properties, such as closure under countable unions and complements. Another approach is to start with a set of generators, which are subsets that can be used to build all other subsets in the Sigma Algebra.

4. What is the relationship between Sigma Algebras and Borel Sets?

Borel Sets are a special type of Sigma Algebra that is constructed from open sets in a topological space. They are important in probability and measure theory because they form the foundation for defining continuous random variables and probability distributions.

5. How are Sigma Algebras used in probability theory?

Sigma Algebras are used in probability theory to define events and calculate probabilities. They allow us to assign a probability to any subset of a sample space, which is necessary for conducting probabilistic experiments and making predictions.

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