Given a topological space , call the sets in the -algebra generated by the open sets of the Borel sets, and call endowed with this structure a Borel space . So in other words, the Borel sets consist of all open and closed sets and their countable intersections and unions. The Borel functions are measurable functions from a Borel space to a topological space. Observe that a continuous function from a Borel space to a topological space is Borel, but the converse is not necessarily true.
It is clear that if is continuous and is measurable, then is measurable, and in fact, the same is true if is Borel. To see this, note that a measurable function pulls all Borel sets back to measurable sets. This observation will be key to our definition of the Lebesgue integral.
Next, define a simple function to be a complex function on a measurable space with an image of finitely many values from . Denote these values by and denote by . The function can then be rewritten as , which is measurable iff is measurable for all .
In Riemann integration, roughly speaking, the familiar approximating blocks heights can be thought of as given by a simple function. And indeed, the next result says that every nonnegative measurable function has a corresponding increasing sequence of approximating simple functions.
Result 5: For measurable , there exist simple measurable functions such that and .
Proof: We will create a sequence of functions that will be Borel by construction, and we will finish by using the above fact that the composition of a Borel function with a measurable function is measurable. Specifically, the functions we will construct are essentially finer and finer approximations of .
where is the greatest multiple of less than . By construction the functions are Borel, increase in $latex $n$, and tend to . Defining gives us the desired sequence of simple functions.
The last building block we must establish before defining the Lebesgue integral is the concept of a measure. Define a positive measure to be a nonnegative function defined on a -algebra which is countably (and thus, as can be shown relatively easily, finitely) additive and takes on a finite value for at least one subset of . A measure space is a measurable space equipped with a positive measure. Alternatively, a complex measure is a measure that takes on complex values. As examples of measures, consider the map from a set to its cardinality and the unit mass at a point. In the case of Riemann integration, consider the map from an interval in to its length!
We can immediately verify that i) , ii) . Additionally, we can show some useful things about the convergence of measures:
Result 6: If where , then .
Proof: This holds for formal reasons. Consider the different sets and for . Then is just the sum of the measures of the first $n$ difference sets, and likewise, is the sum of all difference sets, so indeed tends towards $\latex \mu(A)$.
Result 7: If where , then .
Proof: Again, this holds for formal reasons. Take the differences not between successive sets but between sets and the first set. So let . The union of these difference sets is certainly , and these difference sets are increasing, so by Result 6, we are done.
Okay fine, I lied. Result 7 is only true if there some such that for all , . Use the example of and to convince yourself that this is the case.
Phew, after all of these preliminaries, we are finally ready to get to the interesting stuff, Lebesgue integration for nonnegative and complex functions.