Question

On page 48 of Calculus on Manifolds Spivak defines (Riemann) integration over rectangles $[a_{1},b_{1}]\times\cdots\times[a_{n},b_{n}]\subset\mathbb{R}^{n}$. Then on page 55 he extends this integral to bounded subsets $C\subset\mathbb{R}^{n}$ via characteristic functions. This integral is the usual one.

Then, on page 63 he defines (smooth) partitions of unity and uses them later on page 65 to define and extended integral over open sets $A\subset\mathbb{R}^{n}$.

The usual and extended integrals are not always the same. However, Theorem 3-12 (3) gives us a precise relation between the extended integral and the usual one.

Now, mostly via problems, Spivak makes the reader verify all the familiar properties of the usual integral (linearity, comparison, monotonicity, etc). However, there is no mention in either the theory or the exercises of whether these properties hold for the extended integral. Moreover, when doing the problems I found myself making use of them, so it is natural to ask if the extended integral also verifies these properties. That is:

Let $A$ be an open subset of $\mathbb{R}^{n}$ and $f,g:A\rightarrow\mathbb{R}$ be continuous functions:

1. Linearity: If $f,g$ integrable over $A$, so is $af+bg$ and $\int_{A}af+bg=a\int_{A}f+b\int_{A}g$.
2. Comparison: $f,g$ integrable over $A$ and $f(x)\leq g(x)$ then $\int_{A}f\leq\int_{A}g$. In particular $\left|\int_{A}f\right|\leq\int_{A}|f|$.
3. Monotonicity: If $B \subset A$ is open and $f$ in non-negative on $A$ and integrable over $A$ then it is integrale over $B$ and $\int_{B}f\leq\int_{A}f$.
4. Additivity: If $A$ and $B$ are open and $f$ is continuous on $A\cup B$ and integrable over $A$ and $B$ then it is integrable over the union and the inetrsection and $\int_{A\cup B}f=\int_{A}f+\int_{B}f-\int_{A\cap B}f$.
5. Let $A$ be open and of measure $0$. If $f$ is integrable over $A$ then $\int_{A}f=0$.
6. If $f$ and $g$ agree except on a set of measure $0$ then $\int_{A}f=\int_{A}g$.

I have been verifying these properties and seem to have a proof for each. But I would appreciate it if someone with more experience could corroborate that these properties do indeed hold for extended integrals.

Answer 1

Following the notation in the book (see page 65), if it exists, the extended integral is defined as $\int_{A}f:=\sum_{\varphi\in\Phi}\int_{A}\varphi\cdot f$, where each integral $\int_{A}\phi\cdot f$ exists and being of the the usual type it verifies all the corresponding properties for usual integrals. This and a bit of care, as we are dealing with series, is enough to show each and every one of the properties mentioned above for extended integrals.