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

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**Measures**

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

AnNote 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.

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

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

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

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**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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**The Monotone Convergence Theorem**

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**Fatou's Lemma**

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**The Dominated Convergence Theorem**

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

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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