# Smallest Sigma Algebra .... Axler, Example 2.28 ....

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In summary, the author is trying to figure out why $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$.
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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 make a meaningful start on verifying the first part of Axler, Example 28 ...

The relevant text reads as follows:

Can someone please help me to make a meaningful start on verifying Example 2,28 ... that is, to show that the smallest $\sigma$-algebra on $X$ containing $\mathcal{A}$ is the set of all subsets $E$ of $X$ such that $E$ is countable or $X \setminus E$ is countable ... ...
Help will be much appreciated ...

Peter

Peter said:
Can someone please help me to make a meaningful start on verifying Example 2,28 ... that is, to show that the smallest $\sigma$-algebra on $X$ containing $\mathcal{A}$ is the set of all subsets $E$ of $X$ such that $E$ is countable or $X \setminus E$ is countable ... ...
Let $\mathcal{S}$ be be set of all subsets of $X$ that are either countable or co-countable (where "countable" is understood to include finite or empty, and "co-countable" means having a countable complement). Then $\mathcal{S}$ is a $\sigma$-algebra. To see that, notice that it certainly contains the empty set and is closed under complementation. Also every subset of a countable set is countable, and (by complementation) every superset of a co-countable set is co-countable. Now suppose that $E_1,E_2,\ldots$ is a sequence of sets in $\mathcal{S}$. If anyone of those sets is co-countable then so is the union $\bigcup E_n$. Alternatively, if they are all countable then so is the union. That shows that $\mathcal{S}$ is closed under countable unions and is therefore a $\sigma$-algebra.

But it is easy to see that any $\sigma$-algebra $\mathcal{S}$ that contains $\mathcal{A}$ must contain all the countable subsets of $X$ and their complements. So it contains $\mathcal{S}$. Therefore $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$.

Opalg said:
Let $\mathcal{S}$ be be set of all subsets of $X$ that are either countable or co-countable (where "countable" is understood to include finite or empty, and "co-countable" means having a countable complement). Then $\mathcal{S}$ is a $\sigma$-algebra. To see that, notice that it certainly contains the empty set and is closed under complementation. Also every subset of a countable set is countable, and (by complementation) every superset of a co-countable set is co-countable. Now suppose that $E_1,E_2,\ldots$ is a sequence of sets in $\mathcal{S}$. If anyone of those sets is co-countable then so is the union $\bigcup E_n$. Alternatively, if they are all countable then so is the union. That shows that $\mathcal{S}$ is closed under countable unions and is therefore a $\sigma$-algebra.

But it is easy to see that any $\sigma$-algebra $\mathcal{S}$ that contains $\mathcal{A}$ must contain all the countable subsets of $X$ and their complements. So it contains $\mathcal{S}$. Therefore $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$.
Thanks Opalg ...

Still reflecting on what you have written ...

Peter

Opalg said:
Let $\mathcal{S}$ be be set of all subsets of $X$ that are either countable or co-countable (where "countable" is understood to include finite or empty, and "co-countable" means having a countable complement). Then $\mathcal{S}$ is a $\sigma$-algebra. To see that, notice that it certainly contains the empty set and is closed under complementation. Also every subset of a countable set is countable, and (by complementation) every superset of a co-countable set is co-countable. Now suppose that $E_1,E_2,\ldots$ is a sequence of sets in $\mathcal{S}$. If anyone of those sets is co-countable then so is the union $\bigcup E_n$. Alternatively, if they are all countable then so is the union. That shows that $\mathcal{S}$ is closed under countable unions and is therefore a $\sigma$-algebra.

But it is easy to see that any $\sigma$-algebra $\mathcal{S}$ that contains $\mathcal{A}$ must contain all the countable subsets of $X$ and their complements. So it contains $\mathcal{S}$. Therefore $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$.
Thanks again, Opalg ...

You write:

" ... ... But it is easy to see that any $\sigma$-algebra $\mathcal{S}$ that contains $\mathcal{A}$ must contain all the countable subsets of $X$ and their complements. So it contains $\mathcal{S}$. Therefore $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$. ... "

I am finding it difficult to see exactly why $\mathcal{S}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$. ...

Can you elaborate/explain further ... ?

Peter

The structure of the argument consists of these two parts:
1) $\mathcal{S}$ is a $\sigma$-algebra containing $\mathcal{A}$;
2) Every $\sigma$-algebra that contains $\mathcal{A}$ must contain $\mathcal{S}$.
Those two facts together say that $\mathcal{S}$ is the smallest $\sigma$-algebra containing $\mathcal{A}$.

Thanks Opalg ...

Peter

## 1. What is a Sigma Algebra?

A Sigma Algebra is a collection of subsets of a given set that satisfies certain properties. It is a fundamental concept in measure theory and probability theory, as it allows for the definition of measures and probability measures on a set.

## 2. What is the smallest Sigma Algebra?

The smallest Sigma Algebra, also known as the trivial Sigma Algebra, is the collection of all subsets of a given set. It contains the empty set and the entire set, and is always a Sigma Algebra.

## 3. How is the smallest Sigma Algebra related to Example 2.28 in Axler's book?

In Example 2.28, Axler uses the smallest Sigma Algebra to define the Borel Sigma Algebra, which is the smallest Sigma Algebra containing all open sets in a given topological space. This is a key concept in measure theory and functional analysis.

## 4. Why is the smallest Sigma Algebra important?

The smallest Sigma Algebra is important because it serves as the foundation for the development of more complex Sigma Algebras. It also allows for the definition of measures and probability measures, which are essential in many areas of mathematics and science.

## 5. How is the smallest Sigma Algebra used in real-world applications?

The smallest Sigma Algebra is used in a wide range of real-world applications, including probability theory, statistics, and data analysis. It is also used in fields such as physics, economics, and engineering to model and analyze complex systems and phenomena.

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