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#1 2016-05-29 03:55:05

zetafunc
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Measure Theory

NOTE: These posts are based on David Larman's notes on measure theory, with some examples and exposition added by me. Therefore, some definitions may be non-standard.

One of the aims of measure theory is to explain precisely how one can "measure" a set. We are already familiar with some basic notions: length, area, volume, and probabilities are also examples we've seen before. Can this idea be generalised? That is, can we formulate a method of determining the measure of an arbitrary set? We will eventually define the Lebesgue measure,
, by developing some "nice" properties that one might expect a measure to have. (You may have seen something like this when defining the notion of a metric space.)

Obviously, one might expect
But what about something like
? Since
is "small", you might expect the answer to be 0. These expectations are indeed true. We  also have functions like this:

If you've done any elementary real analysis, you will have seen that such a function is not Riemann integrable, because the upper and lower Riemann integrals are 1 and 0, respectively (they're not equal). However, when we define the Lebesgue integral, we will find that:

Before elaborating in further detail, we'll need to explore some of the fundamentals first. We let
be a set, and
be the family of subsets of
. Then
is an algebra if it satisfies:

,

,

.

For instance, if
, and
is the family of all subsets of
which can be expressed as a finite union of intervals, then
is an algebra.

We say that
is a
-algebra if it is an algebra, but with the added condition:

Notice that, for any
, we have that
is a
-algebra. Furthermore, the example we mentioned above is an algebra, but not a
-algebra.

Last edited by zetafunc (2016-05-29 21:05:34)

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#2 2016-05-29 19:41:38

zetafunc
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Registered: 2014-05-21
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Re: Measure Theory

Measures

Let's talk a little bit about measures. If
is a
-algebra on a set
, then a measure on
is a function
which satisfies:

(i)

(ii) If
are pairwise disjoint, then
This property is called
-additivity.

Examples of Measures

We'll define the Lebesgue measure a little later, as it is perhaps the most important measure pertaining to our discussion. Here are some simple examples of measures -- the reader is invited to verify that these are indeed measures as an exercise.

The Dirac Measure

Fix
, and let
. Define:

The Counting Measure

. This counts the number of elements in E.

The Generalised Counting Measure

assign a number
, and define

Last edited by zetafunc (2016-05-29 20:09:04)

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#3 2016-05-29 20:13:36

zetafunc
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Re: Measure Theory

Properties of Measures

Now that we've explicitly defined a measure, and given some elementary examples of measures, we'll look at some properties that can be deduced about them. These results are quite important, as they characterise some of the "nice" properties we'd want a measure to have.

Last edited by zetafunc (2016-05-29 20:40:34)

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#4 2016-05-30 03:44:32

zetafunc
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Re: Measure Theory

Before we define the Lebesgue measure, we first define the notion of an outer measure.

An outer measure on
is a function
such that:

Note the key differences between this and a measure defined in post #2. In particular, the condition that the sets be pairwise disjoint has been relaxed. Now let's define what it means for a set to be measurable.

Suppose
is an outer measure on
. We say that
is
-measurable if for every
, we have that

We are now ready to define the Lebesgue measure. Let
For any
define the Lebesgue outer measure
by:

Then
is the restriction of
to
-measurable sets. We call
the Lebesgue measure on
We can then deduce that
and moreover the Lebesgue measure of a set containing one element (a singleton) is 0, i.e.

It is possible to construct a subset of
which is not Lebesgue-measurable.

Last edited by zetafunc (2016-05-30 03:47:00)

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#5 2016-05-30 04:02:40

zetafunc
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Re: Measure Theory

Before defining the Lebesgue integral, we first cover some terminology.

A measure space is a triple
, where
is a set,
is a
-algebra on
, and
is a measure on
.

A function
is said to be measurable if
we have

A function is simple if it only takes finitely many values. As you might expect, if
are measurable functions, and
is a sequence of measurable functions, then the following are also measurable:

We'll often need to use indicator functions (or characteristic functions), which take the value of 1 (or 0) depending on whether an element lies in the set (or doesn't). The notation we'll use is
for the characteristic function of a set
.

Last edited by zetafunc (2016-05-30 20:32:49)

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#6 2016-05-30 20:53:20

zetafunc
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Re: Measure Theory

The Lebesgue Integral

We define the integral in three steps.

1) For a simple, non-negative, measurable function
, say,
we define the Lebesgue integral of f with respect to a measure
by:

2) For a non-negative, measurable function
we define:

3) For a measurable function
define:

Many of the familiar properties of the Riemann integral also hold true for the Lebesgue integral. For instance, if
are measurable functions,
, then:

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#7 2016-05-31 20:36:31

zetafunc
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Re: Measure Theory

The Monotone Convergence Theorem

If
is a monotonic increasing sequence of functions converging to
, then:

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#8 2016-05-31 20:39:33

zetafunc
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Re: Measure Theory

Fatou's Lemma

If
are measurable, then:

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#9 2016-05-31 20:41:53

zetafunc
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Re: Measure Theory

The Dominated Convergence Theorem

Suppose
are measurable,
pointwise, and there exists some integrable function
such that for all
, we have
Then:

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#10 2016-06-01 01:37:56

Agnishom
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Registered: 2011-01-29
Posts: 24,833
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Re: Measure Theory

Thanks for the notes. I have not read them all yet, but they seem interesting.


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