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:]]>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.]]>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.
]]>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 ]]>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.]]>