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

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## 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: $$\bar{s}_{f}=\bar{s}_{f+}-\bar{s}_{f-}$$.

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