We will elaborate on a lemma in the book The Elements of Integration and Lebesgue Measure.
Lemma: If f is a nonnegative function in M(X,X), then there exists a sequence () in M(X,X) such that:
(a) for .
(b) for each .
(c) Each has only a finite number of real values.
Let n be a fixed natural number. If k=0, 1, 2, …, , let be the set
If , let .
We note that the sets are disjoint.
The sets also belong to X, and have union equal to X.
Thus, if we define on , then belongs to M(X,X).
We can see that the properties (a), (b), (c) hold.
(a): is true.
(I just noticed there is some typo in Bartle’s book, as the above inequality does not hold. I think n is supposed to be fixed, while k is increased instead.)
(b): As n tends to infinity, on , i.e. , thus for each .
(c): Clearly true!
See also: Recommended Undergraduate Books