A nonnegative function f in M(X,X) is the limit of a monotone increasing sequence in M(X,X)

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 (\phi_n) in M(X,X) such that:

(a) 0\leq \phi_n (x) \leq \phi_{n+1} (x) for x\in X, n\in\mathbb{N}.

(b) f(x) =\lim \phi_n (x) for each x\in X.

(c) Each \phi_n has only a finite number of real values.

Proof:

Let n be a fixed natural number. If k=0, 1, 2, …, n 2^n -1, let E_{kn} be the set

E_{kn}=\{ x\in X: k2^{-n} \leq f(x)<(k+1)2^{-n}\}.

If k=n2^n, let E_{kn}=\{x\in X: f(x) \geq n\}.

We note that the sets \{E_{kn}: k=0, 1,\ldots, n2^n\} are disjoint.

The sets also belong to X, and have union equal to X.

Thus, if we define \phi_n= k2^{-n} on E_{kn}, then \phi_n belongs to M(X,X).

We can see that the properties (a), (b), (c) hold.

(a): 0\leq k2^{-n}\leq k2^{-n-1} 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 k2^{-n} \leq f(x) <(k+1)2^{-n}, i.e. \phi_n (x) \leq f(x) < \phi_n (x)+2^{-n}, thus f(x)=\lim \phi_n (x) for each x\in X.

(c): Clearly true!

Source:

See also: Recommended Undergraduate Books

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