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### 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.

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

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/
```
Associated Topics:
College Analysis
College Modern Algebra

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