My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree1Thanks
  • 1 Post By Maschke
Reply
 
LinkBack Thread Tools Display Modes
August 29th, 2017, 08:13 PM   #1
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 103
Thanks: 0

Exclamation Show the the intersection of two sigma fields is a sigma field

In probability, a sigma field (algebra) is defined as a set F whose elements are sets and that satisfies the following properties:

1. The null set and the sample space are members of F.
2. If an event A is a member of F, then the complement of A is a member of F
3. If events A1, A2, … An are members of F, then the event A1 U A2 … U An is a member of F.

So I have to show that the intersection of ANY two sigma fields is a sigma field. The first property is easy to show. By definition, any any two sigma fields will contain the null set and the sample space, so the intersection between any two sigma fields will inevitably include both the null set and the sample space. Done.

The second two properties are where I'm getting tripped up. I'm not sure how I would show, with mathematical notation, that for the intersection of any two sigma fields in which the resulting set is different from the trivial sigma field ({Null Set, Sample Space}) will always contain at least one event and its complement. I understand it in my head, but don't know how to write it down.

Finally, I have no idea where to start with the proof for the unions. Would somebody please offer some advice on how to write this proof in mathematical notation? Thank you.
John Travolski is offline  
 
August 29th, 2017, 08:22 PM   #2
Senior Member
 
Joined: Aug 2012

Posts: 1,641
Thanks: 415

Doesn't the "sigma" in sigma field indicate that the unions and intersections may be countable? (Yes). Your notation has them as finite. The proof won't be much different either way but if you intend countability you should indicate that.

If X is in the intersection then it's in each of the fields hence its complement is in each field hence ...

For the unions, same kind of reasoning. If X and Y are in the intersection then X and Y are each in both fields therefore X union Y is in each field therefore ...
Thanks from John Travolski

Last edited by Maschke; August 29th, 2017 at 08:57 PM.
Maschke is online now  
August 29th, 2017, 09:04 PM   #3
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 103
Thanks: 0

Okay, I greatly appreciate the help. I believe that I fully understand where to go from here. Thank you.
John Travolski is offline  
September 3rd, 2017, 05:36 AM   #4
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 2,829
Thanks: 753

A set is in the intersection of fields F and G if and only if it is in both F and G. Since the set is in F, its complement is in F. Since it is in G, its complement is in G. Therefore, its complement is in the intersection of F and G.

Similarly for (3). If all of sets A1, A2, ... are in F intersection G then each is in F and in G. Since they are all in F, their union is in F. Since they are all in G, their union is G. Therefor their union is in the intersection of F and G.

Do you see why, if you take the union of F and G it is not necessarily a field?
Country Boy is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
field, fields, intersection, show, sigma



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
By integrating show that EX = muu and DX = sigma rain Probability and Statistics 1 March 28th, 2015 11:17 AM
More sigma... Keroro Algebra 4 June 10th, 2012 07:32 PM
how to find : sigma f/sigma x luketapis Calculus 2 March 27th, 2012 12:30 PM
Sigma Field Question NightBlues Advanced Statistics 0 September 6th, 2010 06:54 PM
show Sigma Phi(d)[n/d] = n(n+1)/2 for n>0 Jamers328 Number Theory 1 December 5th, 2007 10:45 AM





Copyright © 2017 My Math Forum. All rights reserved.