8.5.E: Problems on Integration of Extended-Real Functions

Exercise $\PageIndex{1}$

Using the formulas in ( 1) and our conventions, verify that
(i) $\overline{\int}_{A} f=+\infty$ iff $\overline{\int}_{A} f^{+}=\infty$;
(ii) $\underline{\int}_{A} f=\infty$ iff $\underline{\int}_{A} f^{+}=\infty ;$ and
(iii) $\overline{f}_{A} f=-\infty$ iff $\underline{\int}_{A} f^{-}=\infty$ and $\overline{\int}_{A} f^{+}<\infty$.
(iv) Derive a condition similar to (iii) for $\underline{\int}_{A} f=-\infty$.
(v) Review Problem 6 of Chapter 4, §4.

Exercise $\PageIndex{2}$

Fill in the missing proof details in Theorems 1 to 3 and Lemmas 1 and 2.

Exercise $\PageIndex{3}$

Prove that if $\underline{\int_{A}} f=\infty,$ there is an elementary and (extended) real map $g \leq f$ on $A,$ with $\int_{A} g=\infty$.
[Outline: By Problem $1,$ we have

$$\underline{\int_{A}} f^{+}=\infty .$$
As Lemmas 1 and 2 surely hold for nonnegative functions, fix a measurable $F \leq f^{+}$ $(F \geq 0),$ with
$$\int_{A} F=\underline{\int_{A}} f^{+}=\infty .$$
Arguing as in Theorem $3,$ find an elementary and nonnegative map $g \leq F,$ with
$$(1+\varepsilon) \int_{A} g=\int_{A} F=\infty ;$$
so $\int_{A} g=\infty$ and $0 \leq g \leq F \leq f^{+}$ on $A$.
Let
$$A_{+}=A(F>0) \in \mathcal{M}$$
and
$$A_{0}=A(F=0) \in \mathcal{M}$$
(Theorem 1 in §2). On $A_{+},$
$$g \leq F \leq f^{+}=f(\text { why? }) ,$$
while on $A_{0}, g=F=0 ;$ so
$$\int_{A_{+}} g=\int_{A} g=\infty(\mathrm{why} ?) .$$
Now redefine $g=-\infty$ on $A_{0}$ (only). Show that $g$ is then the required function.]

Exercise $\PageIndex{4}$

For any $f: S \rightarrow E^{*},$ prove the following.
(a) If $\overline{\int}_{A} f<\infty,$ then $f<\infty$ a.e. on $A$.
(b) If $\underline{\int_{A}} f$ is orthodox and $>-\infty,$ then $f>-\infty$ a.e. on $A$.
[Hint: Use Problem 1 and apply Corollary 1 to $f^{+} ;$ thus prove (a). Then for (b), use Theorem 1(e').]

Exercise $\PageIndex{5}$

$\Rightarrow 5$. For any $f, g: S \rightarrow E^{*},$ prove that
(i) $\overline{\int}_{A} f+\overline{\int}_{A} g \geq \overline{\int}_{A}(f+g),$ and
(ii) $\underline{\int}_{A}(f+g) \geq \underline{\int}_{A} f+\underline{\int}_{A} g \quad$ if $\left|\underline{\int}_{A} g\right|<\infty$.
[Hint: Suppose that

$$\overline{\int}_{A} f+\overline{\int}_{A} g<\overline{\int}_{A}(f+g) .$$
Then there are numbers
$$u>\overline{\int}_{A} f \text { and } v>\overline{\int}_{A} g ,$$
with
$$u+v \leq overline{\int}_{A}(f+g) .$$
(Why?) Thus Lemma 1 yields elementary and (extended) real maps $F \geq f$ and $G \geq g$ such that
$$u>\overline{\int}_{A} F \text { and } v>\overline{\int}_{A} G$$
As $f+g \leq F+G$ on $A,$ Theorem $1(\mathrm{c})$ of §5 and Problem 6 of §4 show that
$$\overline{\int}_{A}(f+g) \leq \int_{A}(F+G)=\int_{A} F+\int_{A} G<u+v ,$$
contrary to
$$u+v \leq \overline{\int}_{A}(f+g) .$$
Similarly prove clause (ii).]

Exercise $\PageIndex{6}$

Continuing Problem $5,$ prove that

$$\overline{\int}_{A}(f+g) \geq \overline{\int}_{A} f+\underline{\int}_{A} g \geq \underline{\int}_{A}(f+g) \geq \underline{\int}_{A} f+\underline{\int}_{A} g$$
provided $\left|\underline{\int}_{A} g\right|<\infty$.
[Hint for the second inequality: We may assume that
$$\overline{\int}_{A}(f+g)<\infty \text { and } \overline{\int}_{A} f>-\infty .$$
(Why?) Apply Problems 5 and $4(\mathrm{a})$ to
$$\overline{\int}_{A}((f+g)+(-g)) .$$
Use Theorem $\left.1\left(\mathrm{e}^{\prime}\right) .\right]$

Exercise $\PageIndex{7}$

Prove the following.
(i)

$$\overline{\int}_{A}|f|<\infty \text { iff }-\infty<\underline{\int}_{A} f \leq \overline{\int}_{A} f<\infty .$$
(ii) If $\overline{f}_{A}|f|<\infty$ and $\overline{\int}_{A}|g|<\infty,$ then
$$\left|\overline{\int}_{A} f-\overline{\int}_{A} g\right| \leq \overline{\int}_{A}|f-g|$$
and
$$\left|\underline{\int}_{A} f-\underline{\int}_{A} g\right| \leq \overline{\int}_{A}|f-g| .$$
[Hint: Use Problems $5 \text { and } 6 .]$

Exercise $\PageIndex{8}$

Show that any signed measure $\left.\bar{s}_{f} \text { (Note } 4\right)$ is the difference of two measures: $\bar{s}_{f}=\bar{s}_{f+}-\bar{s}_{f-}$.