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.

Example:

If X is the set** R **of real numbers, and **X** is the Borel algebra **B**, then any monotone function is Borel measurable.

Proof:

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!