Countable Partition Generated ?-Algebra – Measure Theory

elementary-set-theorymeasure-theory

I am working on exercise 1.9 b) in the book "Probability and Stochastics" by Erhan Çınlar to practice my understanding of $\sigma$-algebras (although this might possibly be future homework in my classes).

Anyway, here's the exercise:

Let $\mathcal{C}$ be a (countable) partition of E. Show that every element of $\sigma \mathcal{C}$ is a countable union of elements taken from $\mathcal{C}$.

So, $\mathcal{C} = \{ C_n: n \in \mathbb{N}\}$ is a partition of $E$.
To prove the exercise, I have introduced $\mathcal{E} = \{\cup_{i \in I} C_i: I \subseteq \mathbb{N}\}$ and am trying to show that $\mathcal{E}$ is a $\sigma$-algebra.

For this, I have to show the following three parts/properties (correct me if I'm wrong):

  1. E $\in \mathcal{C}$: $\mathcal{C}$ is a partition of E, so $\cup \mathcal{C}=E.$ Since $\mathcal{C}$ is countable, $\cup \mathcal{C}$ is the union of countably many members of $\mathcal{C}$.

  2. Show that $\mathcal{E}$ is closed under countable unions: Since $\mathcal{E}$ consists only of sets which are countable unions of elements of $\mathcal{C}$, $\mathcal{E}$ is closed under countable unions.

  3. Show that $\mathcal{E}$ is closed under complement: Let $A \in \mathcal{C}$, so there exists a countable $\mathcal{C}_A \subseteq \mathcal{C}$ such that $A = \cup \mathcal{C}_A.$ Let $\mathcal{D} = \mathcal{C} \setminus \mathcal{C}_A$; then $\mathcal{D}$ is a countable subset of $\mathcal{C}$, and if $D =\cup \mathcal{D},$ then $D \in \mathcal{C}.$ Since $\mathcal{C}$ is a partition on E, $D = E \setminus A = A^c$, and so $A^c\in \mathcal{C}$.

Now, in the last part I want to show that $\mathcal{E} = \sigma\mathcal{C}$, but I'm not sure how exactly this is done.

Best Answer

Here's a proof using only the material covered in Çınlar before your exercise. As you showed, $\mathcal{E}$ is a $\sigma$-algebra containing $\mathcal{C}$. It is obviously contained in $\sigma(\mathcal{C})$ : to see that, all we have to say is that $\sigma(\mathcal{C}$ must contain all the countable unions of elements in $\mathcal{C}$, so it contains all the elements of $\mathcal{E}$. Therefore, $\mathcal{E} \subset \sigma (\mathcal{C})$.

But, $\sigma(\mathcal{C})$ is defined as the intersection of all the $\sigma$-algebras containing $\mathcal{C}$. Therefore, as $\mathcal{E}$ is a $\sigma$-algebra containing $\mathcal{C}$, it is clear that $\sigma(\mathcal{C})$ is contained in $\mathcal{E}$.

So, $\mathcal{E} = \sigma (\mathcal{C})$.

This kind of reasonning is very very common in measure theory, analysis, probability. Make sure you understand all the things here. It works like this : to prove that two $\sigma$-algebras are equal, you write one as the $\sigma$-algebra generated by a collection, and you show that the other is a $\sigma$-algebra containing this collection and contained in the first one.

Have a nice time with Çınlar's excellent textbook :)

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