Measure Theory and Sigma Algebras

Date: 03/24/2003 at 22:09:19
From: Trevor
Subject: Measure Theory

I'm trying to understand what a "measure" is. I have definitions, but 
no explanations to help me.

What is confusing is all the emphasis on the domain being a sigma 
algebra. We work this all up, add three conditions and "Measure."  
WHAT? what can I measure?

A measure is defined (in my own words) as an extended non-negative-
valued function whose domain is a sigma-field (or sigma-algebra) of 
subsets of some set that satisfies countable subadditivity, countable 
additivity, and the empty set is equal to zero.

Why do I want the term "measure" to be equal to that definition? What 
am I trying to do here?

Help?

Date: 03/25/2003 at 00:32:00
From: Doctor Tom
Subject: Re: Measure Theory

Hi Trevor,

Let me try to give you an idea of why this definition of measure is 
"good" in a special situation. Let's look at the problem of trying to 
assign probabilities to a situation, and let's consider the situation 
where the outcome of the experiment is a real number.

We'd like the total probability (total measure) to be 1.

If we roll a die, there are precisely 6 outcomes; it can be 1, 2, 3, 
4, 5 or 6.  It cannot be 1.3, or 2.7, or 7 or -12.4. So for each point 
on the line, we assign probability of 1/6 (assuming it's a fair die) 
and assign 0 to every other event.

I can ask, "What's the probability of the result being between 1.3 and 
4.7?"  Obviously, it's 1/2 - it could be 2 or 3 or 4 only. But even 
though the die can only give integral results, weird questions like 
the one above DO have an answer.

Now suppose the experiment is different - a dart is thrown into the 
inteval between 0 and 1 and it is equally likely to hit any point in 
that region.  Clearly the probability of hitting any particular point 
is zero (since there is an infinite number of them) but there is some 
probability that can be assigned to intervals or combinations of 
intervals, or in fact to sigma-algebras of intervals that will be 
totally reasonable. In fact, if you know that for any interval [a, b] 
where 0 < a < b < 1 the probability of the dart landing there is b-a.  
And from this, you can assign in a reasonable way measures to any set 
in the sigma-algebra generated by intervals.

Now suppose the dart is thrown at the real line with the target at 
zero, but the thrower is not perfect. Then you can again assign 
probabilities to the dart falling in any interval (presumably higher 
for intervals near zero than for equal-sized intervals far away), and 
if you can assign such probabilites for intervals, you can assign them 
for any set in the sigma algebra generated by those intervals.

Now how about this: You're tossing darts at a target on the wall, and 
you want to measure the height above the floor that the dart ends up.  
For any exact height in the target, the probability is zero, but for 
any interval of heights, you can assign a measure. But there's also a 
chance that the dart doesn't stick and falls to the floor.  ALL of 
those darts end at height zero. So the probabiliy of the point 0 is 
not zero - it is some finite number. This kind of distribution assigns 
a continuous measure to part of the situation (on the board) and 
discrete to the floor (like dice). There could, of course, be multiple 
discrete values, like if the board is over a set of stairs - there is 
a different non-zero probability that the dart could wind up at the 
particular height of each step.

Now, the reason sigma algebras are interesting is that they, in some 
sense, are the LARGEST collection of sets to which a reasonable 
measure can be assigned. I don't know if you've gotten to them yet, 
but there are sets that are not measurable in the sense that they are 
perfectly reasonable sets (or at least people who belive in the axiom 
of choice think so), but there is no sane way to assign a measure to 
them that isn't somehow contradictory.

The discovery of non-measurable sets was a giant surprise, since the 
first investigators of the problem of assigning measures to, say, the 
real line, assumed that they could assign a reasonable measure to 
EVERY subset, and they were wrong.  So they found some bad subsets, 
and the whole theory of sigma-algebras was created to be able to 
categorize exactly which subsets could be measured (all of the ones in 
the sigma algebra) and which ones could not (any set NOT in the sigma 
algebra).

I hope this helps a little.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/