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/
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