$\sigma$-algebra proof

Question

I am preparing for a probability exam and i am stuck with the following exercise from an old exam.

Given is the probability space $(\Omega,\Gamma ,P )$ with $\boldsymbol{A}:=\left \{ A\in \Gamma: P(A) \in \left \{ 0,1 \right \} \right \}$. a) Show that $\boldsymbol{A}$ is a sigma algebra.

b) Give an example of $\boldsymbol{A}=\left \{ \varnothing , \Omega \right \}$ with $\boldsymbol{A}$ containing endless many elements.

With (a) I started the following way with proving the properties of a sigma algebra:

(1) $\Omega \in \boldsymbol{A}$ . Since $\varnothing \in \Omega$, then $P(\varnothing)=0 \in \boldsymbol{A}$

(2) If $A\in \boldsymbol{A}$, then $A^{c}\in \boldsymbol{A}$. If $P(A)=0$, then $P(A^{c})=1\in \boldsymbol{A}$. And $P(A)=1$, so $P(A^{c})=0\in \boldsymbol{A}$.

(3) Here i have a problem with showing that if $A:=\bigcup A_{n}, n\in \mathbb{N}$, then $A\in \boldsymbol{A}$

With finding an example for (b) i have problems too.

I would be very glad if anyone could help me with this exercise. Thank you in advance!

Answer 1

Ad 3):

Suppose you have a sequence $A_n$ of events such that $\mathbb{P}(A_n)=1$ or $\mathbb{P}(A_n)=0$ for every $n$. There are two cases: all $A_n$ have probability $0$, or some $A_n$ have probability $1$.

In the first case we have $\mathbb{P}(\bigcup A_n)\leq \sum \mathbb{P}(A_n)=0$, and therefore $\mathbb{P}(\bigcup A_n)=0$ and $\bigcup A_n\in \mathcal{A}$.

In the second case we can assume that $\mathbb{P}(A_k)=1$ for some $k$. Then $\mathbb{P}(\bigcup A_n)\geq \mathbb{P}(A_k)=1$ and thus $\mathbb{P}(\bigcup A_n)=1$, so $\bigcup A_n\in \mathcal{A}$.

Now the example you need: consider the probability space $(\mathbb{N},2^\mathbb{N},\mathbb{P})$ where $\mathbb{P}$ is given by $\mathbb{P}(\{k\})=1/2^{(k+1)}$. Note that this probability space has the property you need.

Comment: I think there is a misunderstanding: do you need a probability space for which this sigma algebra is trivial and the space itself is not, or the trivial probability space and trivial algebra (this is trivial:)? My example serves the first requirement. Please edit your question and make it clear. (I assume you want $\Gamma$ being infinite).

Answer 2

(a)(3) There are the following two cases:

• $\mathbb{P}(A_n)=0$ for all $n \in \mathbb{N}$. Use $\mathbb{P}(A) \leq \sum_{n \in \mathbb{N}} \mathbb{P}(A_n)$.
• There exists $n_0 \in \mathbb{N}$ such that $\mathbb{P}(A_{n_0})=1$. Use $\mathbb{P}(A) \geq \mathbb{P}(A_{n_0})$.

(b) Consider for example $\Omega = \mathbb{N}_0$, $\Gamma := 2^{\Omega}$, $\mathbb{P} := \frac{6}{\pi^2} \cdot \sum_{j=0}^{\infty} \frac{1}{j^2} \delta_j$

(Probably there is typo in your question: If $A=\{\emptyset,\Omega\}$, then $A$ has exactly two (and not endless many) elements.)