I would like some help with the following exercise from Bartle's "Elements of Integration and Lebesgue Measure" (exercise 4.D).

I'd appreciate any hints, since I have no idea on how to use the given observation.

4.D. If $f \in M^{+}$ (the set of nonnegative extended real-valued measurable functions on the measure space ($X$,$\boldsymbol{X}$)) and $c > 0$, then the mapping $\phi \to \psi = c\,\phi$ is a one-to-one mapping between simple functions $\phi \in M^{+}$ with $\phi \leq f$ and simple functions $\psi \in M^{+}$ with $\psi \leq cf$. Use this observation to give a different proof of Corollary 4.7(a).

Where Corollary 4.7(a) states:

4.7 Corollary. (a) If $f$ belongs to $M^{+}$ and $c \geq 0$, then $cf$ belongs to $M^{+}$ and

$$ \int cf\,d\mu = c \int f\,d\mu.$$

and the proof given is

Proof. a) If $c = 0$ then the result is immediate. If $c > 0$, let $(\phi_n)$ be a monotone increasing sequence of simple functions in $M^{+}$ converging to $f$ on $X$ (see Lemma 2.11). Then $(c\phi_n)$ is a monotone sequence converging to $cf$. If we apply Lemma 4.3(a) and the Monotone Convergence Theorem, we obtain

$$\begin{align} \int cf\,d\mu &= lim \int c\,\phi_n \,d\mu\\ &= c\:lim\int \phi_n\,d\mu = c \int f\,d\mu.\end{align}$$

  • $\begingroup$ What is lemma 4.3(a) ? $\endgroup$ – StuartMN Oct 9 '17 at 3:54
  • $\begingroup$ @StuartMN Sorry, I thought the other lemmas were irrelevant to the question. 4.3 Lemma. (a) If $\phi$ and $\psi$ are simple functions in $M^{+}$ and $c \geq 0$, then $\int c\phi\,d\mu = c \int \phi\,d\mu$, and $\int (\phi + \psi)\, d\mu = \int \phi\,d\mu\, + \int \psi\,d\mu$. The other lemma mentioned, lemma 2.11, states that if $f \in M^{+}$, then there exists a monotone increasing sequence of simple functions in $M^{+}$ that converges to $f$. $\endgroup$ – mlaci Oct 9 '17 at 4:00

I guess they want you to say that the integral of cf is the sup of the integrals of all non-negative simple functions less of equal to cf which by there remarks is the sup of the set consisting of numbers c times the integral of some simple function less or equal to f which is = to c times the sup of the integrals of simple functions less or equal to f which is c times integral of f . Sorry for all the words but I am too lazy to look up the math symbols right now .Hope this helps

  • $\begingroup$ Yes! I cannot believe I didn't think of this. Thank you! $\endgroup$ – mlaci Oct 9 '17 at 10:20

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