Sigma Fields

Show that a $\sigma$-field is also a field.

If a field $\mathcal{F}$ is finite, then it is also a $\sigma$-field.

Show that the family of all subsets of $\Omega$ is a $\sigma$-field.

Is the family $\mathcal{F}$ consisting of all finite subsets of $\Omega$ and their complements always a $\sigma$-field?

Is the family $\mathcal{F}$ consisting of all countable subsets of $\Omega$ and their complements always a $\sigma$-field?

Let $\mathcal{F}$ be a $\sigma$-field on $\Omega =[0,1]$ such that $\left[\frac{1}{n+1}, \frac{1}{n}\right] \in \mathcal{F}$ for $n= 1,2, \ldots$. Show that

Let $\mathcal{F}$ be a $\sigma$-field. Demonstrate that if $A_{1}, A_{2}, \ldots \in \mathcal{F}$, then

Let $\Omega$ and $\tilde{\Omega }$ be arbitrary sets and let $X: \tilde{\Omega } \rightarrow \Omega$ be any function. Show that if $\mathcal{F}$ is a $\sigma$-field on $\Omega$, then $\tilde{\mathcal{F}}=\left\{ X^{-1}(A): A \in \mathcal{F}\right\}$ is a $\sigma$-field on $\tilde{\Omega }$.

Let $\mathcal{F}$ be a $\sigma$-field on $\Omega$ and let $A \subset \Omega$. Show that $\tilde{\mathcal{F}}=\{ A \cap B$ : $B \in \mathcal{F}\}$ is a $\sigma$-field on $A$.

Let $\tilde{\Omega }$ and $\Omega$ be arbitrary sets and let $X: \tilde{\Omega } \rightarrow \Omega$ be any function. Show that if $\tilde{\mathcal{F}}$ is a $\sigma$-field on $\tilde{\Omega }$, then $\mathcal{F}=\left\{ A \subset \Omega : X^{-1}(A) \in \tilde{\mathcal{F}}\right\}$ is a $\sigma$-field on $\Omega$.

Find an example of a function $X: \tilde{\Omega } \rightarrow \Omega$ and a $\sigma$-field $\tilde{\mathcal{F}}$ on $\tilde{\Omega }$ such that $\{ X(A): A \in \tilde{\mathcal{F}}\}$ is not a $\sigma$-field.

Is the field of periodic subsets of $\mathbb {Z}$ in Problem 3.20 a $\sigma$-field?

Find two $\sigma$-fields such that their union is not a $\sigma$-field.

Prove that if $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $\sigma$-fields on $\Omega$, then $\mathcal{F}_{1} \cap \mathcal{F}_{2}$ is also a $\sigma$-field.

Show that if $\left\{ \mathcal{F}_{j}\right\}_{j \in J}$ is any collection of $\sigma$-fields defined on the same set $\Omega$, then their intersection $\bigcap_{j \in J} \mathcal{F}_{j}$ is also a $\sigma$-field.

Let $\mathcal{A}=\{ [0,1]\} \cup \left\{ \left[\frac{1}{2^{n+1}}, \frac{1}{2^{n}}\right): n=0,1,2, \ldots \right\}$. Which of the following sets belong to $\mathcal{F}_{\mathcal{A}}$:

Show that $\mathcal{F}_{\mathcal{A}}=\mathcal{A}$ if and only if $\mathcal{A}$ is a $\sigma$-field.

Let $\tilde{\mathcal{F}}$ be a $\sigma$-field that satisfies the conditions

1. $\mathcal{A} \subset \tilde{\mathcal{F}}$;

2. if $\mathcal{F}$ is a $\sigma$-field and $\mathcal{A} \subset \mathcal{F}$, then $\tilde{\mathcal{F}} \subset \mathcal{F}$.

Show that $\mathcal{F}_{\mathcal{A}}=\tilde{\mathcal{F}}$.

Let $\mathcal{A}$ be the family of all finite subsets of $\Omega$. Show that $\mathcal{F}_{\mathcal{A}}$ is the $\sigma$-field consisting of all countable subsets of $\Omega$ and their complements introduced in Problem 5.5.

If $\mathcal{C} \subset \mathcal{D}$, then $\mathcal{F}_{\mathcal{C}} \subset \mathcal{F}_{\mathcal{D}}$.

Show that the following are Borel sets (that is, belong to $\mathcal{B}$):

Show that $(-\infty , a],[a,+\infty ),[a, b),(a, b]$, and the set $\mathbb {Z}$ of whole numbers are Borel.

To construct the Cantor set, we remove the open interval $\left(\frac{1}{3}, \frac{2}{3}\right)$ from the interval $[0,1]$. Next, we divide each of the remaining two intervals into three equal subintervals and remove the middle ones, $\left(\frac{1}{9}, \frac{2}{9}\right)$ and $\left(\frac{7}{9}, \frac{8}{9}\right)$. In this manner, we carry on ad infinitum; see Figure 5.1, which shows the first five steps of the construction. The Cantor set is what remains after this infinite process.

Verify that the Cantor set is a Borel set.

: FIGURE 5.1. Construction of the Cantor set

Show that the $\sigma$-field $\mathcal{B}$ of Borel sets coincides with the $\sigma$-field generated by the family of open sets in $\mathbb {R}$.

Show that if $\mathcal{A}$ in Definition 5.3 is replaced by the family $\mathcal{C}$ of closed intervals, then the resulting $\sigma$-field is also $\mathcal{B}$.

Show that the inverse image of a Borel set under a continuous function $f: \mathbb {R} \rightarrow \mathbb {R}$ is Borel.

Let $A_{1}, A_{2}, \ldots \in \mathcal{F}$, where $\mathcal{F}$ is a $\sigma$-field. Show that

Show that

Find $\limsup_{n \rightarrow \infty } A_{n}$ and $\liminf_{n \rightarrow \infty } A_{n}$, where

Show that $\liminf_{n \rightarrow \infty } A_{n} \subset \limsup_{n \rightarrow \infty } A_{n}$.