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.

Proof:

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!

Source:

See also: Recommended Undergraduate Books