Is (-infinity, b) an event for any real number b?

[kolua]

Homework Statement

Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event.

The Attempt at a Solution

use the 3 conditions required for sigma-algebra.
1. S is an event.
2. If A is an event then A^cis also an event
3. if A_a, A₂... is a countable collection of events, the union of such events is an event.

for the first condition, s is the subset of itself, so it's an event
for the second condition, I am not sure how to prove that [b, infinity) is also an event

 

[andrewkirk]

A countable union of events is an event. Can you think of a way of expressing the open interval $(-\infty,b)$ as a countable union of events $\bigcup_{i=1}^\infty (-\infty,a_i]$? How might you choose the $a_i$?

 

[kolua]

A countable union of events is an event. Can you think of a way of expressing the open interval $(-\infty,b)$ as a countable union of events $\bigcup_{i=1}^\infty (-\infty,a_i]$? How might you choose the $a_i$?

Yes, I know how to prove the third condition. a_i=b-1/n as a goes to infinity. But what about the second condition? how should I prove that A^c is an event? A^c=[b, infinity)

 

[Ray Vickson]

Yes, I know how to prove the third condition. a_i=b-1/n as a goes to infinity. But what about the second condition? how should I prove that A^c is an event? A^c=[b, infinity)

For integer $n > 0$ the set $(-\infty,b- \frac{1}{n}]^c = (b-\frac{1}{n},\infty)$ is an event. We have
$$ [b ,\infty) = \bigcap_{n=1}^{\infty} \left(b - \frac{1}{n}, \infty \right).$$

 

[andrewkirk]

Yes, I know how to prove the third condition. a_i=b-1/n as a goes to infinity. But what about the second condition? how should I prove that A^c is an event? A^c=[b, infinity)

You don't have to prove the second condition.

The conditions tell us how to construct the sigma algebra generated by a collection of sets. Given a collection C of sets, the sigma algebra generated by that collection is the smallest collection of sets that (1) contains C and (2) satisfies those three properties.

What you're asked to do is, given that C is the set of intervals $(-\infty,b]$ for $b\in\mathbb R$, show that for any $b\in\mathbb R$, the interval $(-\infty, b)$ is in S, the sigma algebra generated by C.

To do that we only need to use property 3. We already know that $C\subseteq S$. Property 3 shows that it follows from that that for any $b\in\mathbb R$, the interval $(-\infty, b)$ is also in S.