I am reading Analysis on Manifolds by James R. Munkres.
Munkres defines the extended integral of a continuous function as follows:
Definition. Let $A$ be an open set in $\mathbb{R}^n$; let $f:A\to\mathbb{R}$ be a continuous function. If $f$ is non-negative on $A$, we define the (extended) integral of $f$ over $A$, denoted $\int_A f$, to be the supremum of the numbers $\int_D f$, as $D$ ranges over all compact rectifiable subsets of $A$, provided this supremum exists. In this case, we say that $f$ is integrable over $A$ (in the extended sense). More generally, if $f$ is an arbitrary continuous function on $A$, set $$f_{+}(x)=\max\{f(x),0\}\text{ and }f_{-}(x)=\max\{-f(x),0\}.$$ We say that $f$ is integrable over $A$ (in the extended sense) if both $f_{+}$ and $f_{-}$ are; and in this case we set $$\int_A f=\int_A f_{+}-\int_A f_{-},$$ where $\int_A$ denotes the extended integral throughout.
My question is, if $f$ and $g$ are integrable over an open set $A$, does the function $fg$ is integrable over $A$? My intuition says yes and my first attempt was to use the following Theorem:
Let $A$ be open in $\mathbb{R}^n$; let $f : A\rightarrow\mathbb{R}$ be a continuous function. Choose a sequence $C_N$ of compact rectificable subsets of $A$ whose union is $A$ such that $C_N\subseteq\text{int}(C_{N+1})$ for each $N$. Then $f$ is integrable over $A$ if and only if the sequence $\left\{\int_{C_N} |f|\right\}$ is bounded. In this case, $$\int_{A} = \lim_{N\to\infty}\int_{C_N} |f|$$
But I'm not sure how to prove that the sequence $\left\{\int_{C_N} |fg|\right\}$ is bounded (if it is).
NOTE: Rectificable means Jordan-measurable.
