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

• kolua
In summary, the conversation discusses how to show that the interval (-infinity, b) is an event in the sigma-algebra generated by the set of all real numbers and all intervals of the form (-infinity, b]. The three conditions for sigma-algebra are used and it is proven that (-infinity, b) can be expressed as a countable union of events, satisfying the third condition. It is also stated that the second condition does not need to be proven, as it is a property of sigma-algebras.
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 Acis also an event
3. if Aa, 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

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##?

andrewkirk said:
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. ai=b-1/n as a goes to infinity. But what about the second condition? how should I prove that Ac is an event? Ac=[b, infinity)

kolua said:
Yes, I know how to prove the third condition. ai=b-1/n as a goes to infinity. But what about the second condition? how should I prove that Ac is an event? Ac=[b, infinity)

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

Last edited:
kolua said:
Yes, I know how to prove the third condition. ai=b-1/n as a goes to infinity. But what about the second condition? how should I prove that Ac is an event? Ac=[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.

## 1. What is an event in a scientific context?

In science, an event refers to a specific occurrence or happening that can be observed and measured. It can be a singular event or a series of events that follow a certain pattern.

## 2. How do you prove that a set is an event?

In order to prove that a set is an event, you need to demonstrate that it meets the criteria of an event. This includes being observable, measurable, and having a clear beginning and end. You may also need to provide evidence or data to support your claim.

## 3. What are some examples of events in science?

Examples of events in science include natural phenomena such as eclipses, earthquakes, and volcanic eruptions, as well as human-made events such as experiments, chemical reactions, and technological advancements.

## 4. Can a set be both an event and a non-event at the same time?

No, a set cannot be both an event and a non-event simultaneously. An event is something that is observable and measurable, while a non-event is something that does not occur or does not have a clear beginning and end. Therefore, a set can only be classified as one or the other.

## 5. Why is it important to prove that a set is an event in scientific research?

In scientific research, it is crucial to clearly define and identify events in order to accurately collect data and draw conclusions. Proving that a set is an event ensures that it is a valid and reliable part of the research process, leading to more accurate and meaningful results.

Replies
1
Views
1K
Replies
6
Views
2K
Replies
8
Views
1K
Replies
10
Views
1K
Replies
3
Views
1K
Replies
5
Views
1K
Replies
8
Views
4K
Replies
2
Views
3K
Replies
15
Views
2K
Replies
7
Views
2K