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:

*Linearity*: If $f,g$ integrable over $A$, so is $af+bg$ and $\int_{A}af+bg=a\int_{A}f+b\int_{A}g$.*Comparison*: $f,g$ integrable over $A$ and $f(x)\leq g(x)$ then $\int_{A}f\leq\int_{A}g$. In particular $\left|\int_{A}f\right|\leq\int_{A}|f|$.*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 $\int_{B}f\leq\int_{A}f$.*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 $\int_{A\cup B}f=\int_{A}f+\int_{B}f-\int_{A\cap B}f$.- Let $A$ be open and of measure $0$. If $f$ is integrable over $A$ then $\int_{A}f=0$.
- 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.**