This is a continuation of the study of the book The Elements of Integration and Lebesgue Measure by Bartle, listing a few examples of functions that are measurable. Bartle is a very good author, he tries his very best to make this difficult subject accessible to undergraduates.
If X is the set R of real numbers, and X is the Borel algebra B, then any monotone function is Borel measurable.
Suppose that f is monotone increasing, i.e. implies .
Then, consists of a half-line which is either of the form or the form . (We will show later that both cases can occur.) Thus, the set will belong to the Borel algebra B which is the -algebra generated by all open intervals (a,b) in R.
Both cases can indeed occur. For example, if f(x)=x, then the set will be of the form . More interestingly, if the set is the step function , then when , the set will be .
Lemma: An extended real-valued function f is measurable if and only if the sets , belong to X and the real-valued function defined by is measurable.
This lemma is often useful when dealing with extended real-valued functions.
Proof: If f is in M(X,X), it is proven earlier in the book by Bartle that A and B belong to X. Let and , then we have that which is in X since it is the complement of the union of A and .
If , then , which is a union of two sets in X and hence also in X.
Hence, is measurable.
Conversely, if and is measurable, then when , and when , due to a similar reason as above. Therefore f is measurable!