Let $U \subseteq \mathbb{R}^n$ be open(not necessarily bounded), let $f,g: U \rightarrow \mathbb{R}$ be continuous, and suppose that $|f(x)| \leq g(x)$ for all $ x \in U$. Show if $\int_U g$ exists, so does $\int_U f$
Definition: Let $f:S \rightarrow \mathbb{R}$ be a (not necessarily bounded) function on a set $S \subseteq \mathbb{R}^n$ (not necessarily bounded), and suppose $f \geq 0$, define $\int_S f=\textbf{sup} \{\int_D f: D \subseteq S$ $\textbf{is compact and rectifiable} \}$, provided this supremum exists. For general $f$ (which may be negative at some point), define $\int_S f=\int_S f^+-\int_S f^-$, provided both these integrals exist. (Re call: $f^+=\textbf{max}\{f,0 \}, f^-=\textbf{max \{-f,0\}}$). $\int_S f$ is the extended integral over S.
Lemma: Let $ u \subseteq \mathbb{R}^n$ be open. There exists a sequence $\{C_N:N \in \mathbb{N} \}$ of compact, rectifiable sets such that $C_N \subseteq C_{N+1}$ for each $N \in \mathbb{N}$, and $u=\bigcup_{N=1}^{\infty} C_N$.
Theorem: Let $u \subseteq \mathbb{R}^n$ be open, let $f: u \rightarrow \mathbb{R}$ be continuous, and let $\{C_N:N \in \mathbb{N} \}$ be a sequence of compact rectifiable sets. f is integrable over $u$ if and only if $\{\int_{C_N} |f| \}^{\infty}_{N=1}$ is a bounded sequence of regular integrals.
Corollary: Let $fLu \rightarrow \mathbb{R}$ be a continuosu function on an open set $u$. Then $f$ is integrable over $u$ iff $|f|$ is integrable over $u$.
Theorem: let $u \subseteq \mathbb{R}^n$ be open and bounded, and let $f: u \rightarrow \mathbb{R}$ be bounded and continuous. Then the extended integral of $u$ exists. If the regular integral of $f$ also exists, the two are equal.
Question: The above are the tools I have for this one. Yet I am not sure how to use them. We know one already exists. Should I say that the other one is bounded and lead all the way to integrability and hence exists?
Best Answer
Hint:
Since $f$ is continuous, it is integrable over each compact rectifiable set $C_N$ and, since $|f| \leqslant |g|$, we have for all $N \in \mathbb{N}$
$$\tag{*}\int_{C_N} |f| \leqslant \int_{C_N}|g| \leqslant \sup_K\int_{C_K}|g|$$
Apply the first theorem (both directions) -- to conclude that the RHS of (*) is finite and finally that $f$ is integrable over $U$.