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)
MeasuresLet'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 MeasureFix , and let . Define:
The Counting Measure. This counts the number of elements in E.
The Generalised Counting Measureassign a number , and define
Last edited by zetafunc (2016-05-29 20:09:04)
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)
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)
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)
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:
The Dominated Convergence TheoremSuppose are measurable, pointwise, and there exists some integrable function such that for all , we have Then:
Thanks for the notes. I have not read them all yet, but they seem interesting.
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