I'm reading section "Integration of $\overline{\mathbb{R}}$-valued functions" on page 103 from Amann's texbook Analysis III.
The decomposition of an $\overline{\mathbb{R}}$-valued function into its positive and negative parts allows us also to extend the Lebesgue integral to measurable $\overline{\mathbb{R}}$-valued functions admitting negative values. We say that $f \in \mathcal{L}_{0}(X, \mu, \overline{\mathbb{R}})$ is Lebesgue integrable with respect to ${\mu}$ if $\int_{X} f^{+} d \mu<\infty$ and $\int_{X} f^{-} d \mu<\infty$. In this case, $$ \int_{X} f d \mu:=\int_{X} f^{+} d \mu-\int_{X} f^{-} d \mu $$ is called the (Lebesgue) integral over $X$ with respect to the measure $\mu$.
3.11 Remarks: For $f \in \mathcal{L}_{0}(X, \mu, \overline{\mathbb{R}})$, these three statements are equivalent:
- (i) $f$ is Lebesgue integrable with respect to $\mu$.
- (ii) $\int_{X}|f| d \mu<\infty$;
- (iii) There exists $g \in \mathcal{L}_{1}(X, \mu, \mathbb{R})$ such that $|f| \leq g \mu$-a.e.
Proof:
"(i) $\Rightarrow$ (ii)" This is a consequence of $|f|=f^{+}+f^{-}$.
"(ii) $\Rightarrow$ (iii)" Theorem $3.9$ says that $|f| \in \mathcal{L}_{1}(X, \mu, \mathbb{R})$. Hence (iii) holds with $g=|f|$.
"(iii) $\Rightarrow(\mathrm{i})$ " This follows from $f^{+} \vee f^{-} \leq|f| \leq g$ and Remark 3.3(b).
Theorem $3.9$ is as follows:
Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, | \cdot |)$ a Banach space. For $f \in \mathcal{L}_{0}(X, \mu, E)$, the following are equivalent:
(i) $f \in \mathcal{L}_{1}(X, \mu, E)$;
(ii) $|f| \in \mathcal{L}_{1}(X, \mu, \mathbb{R})$;
(iii) $\int_{X} |f| \mathrm d \mu<\infty$ in the sense of Lebesgue integral.
Here $\mathcal L_0 (X, \mu, E)$ is the space of all $\mu$-measurable functions and $\mathcal L_1 (X, \mu, E)$ the space o all $\mu$-integrable functions. Related definitions of Bochner integral whose related definitions can be found here.
My question: My concern is the proof of "(ii) $\Rightarrow$ (iii)".
To apply Theorem 3.9, $E$ has to be a Banach space. Clearly, the extended real line $\overline{\mathbb{R}} := \mathbb R \cup \{\pm \infty\}$ is a topological space with order topology but not a metric space [ref], let alone Banach space. So I feel the use of Theorem 3.9 is not appropriate.
Also, $f$ possibly takes values $\pm \infty$, so it is not necessarily true that $|f| \in \mathcal{L}_{1}(X, \mu, \mathbb{R})$.
Could you elaborate on my confusion?
