On Spivak "Calculus on Manifolds" he builds the concept of integration on an incremental fashion:
-
He starts by defining the integral $\int_R f$ on a rectangle R;
-
Next he define the concept of characteristic function:
\begin{equation}
X_C = 1 \text{ if }x\in C \text{ else } 0.
\end{equation}
And use this concept for generalize the definition of integral for a
region $C$, by defining $\int_C f = \int_R f \cdot X_C$ for $C$
contained in a rectangle $R$. This concept works for all the cases when C
boundary has measure 0 and $X_C$ is integrable (see theorem 3-9 of the
same book). -
Then he defines partitions of the unit to generalize this concept even further. Using the concept of partition of the unit he defines the integral in the extended sense as:
\begin{equation}
\sum_{\phi \in \Phi}\int_A\phi \cdot f
\end{equation}
where $\Phi$ is a collection of functions such that $\phi \in \Phi$. Some properties of this functions are described next
Be $A$ a bonded region and $O$ and open cover to it, it can be proved
that (see theorem 3-11 of the same book) there exist a collection $\Phi$ of $C^\infty$ functions such:
- $0 \le\phi(x) \le 1$
- A finite number of $\phi(x)$ is different than zero in a open set containing $x \in A$
- $\sum_{\phi \in \Phi} \phi(x) = 1$
- For each $\phi \in \Phi$ there is an open set $U \in O$ such that $\phi=0$ outside of some closed set contained in $U$. Let us call this closed set $C$.
So my question is: how can we prove $\int_A\phi \cdot f$ is integrable?
My understanding about the question is the following: From the above definition it follows that $\int_A\phi \cdot f = \int_C\phi \cdot f$. So if $C$ boundary has measure $0$ we could use the previous definition of integration to say this function is integrable in this region…But how can we prove that this is indeed the case?
Best Answer
I studied from Calculus on Manifolds this year, and in this section, I found that his treatment was a little sloppy. First, there is a huge error in the entire section of partitions of unity: in property ($4$) of Theorem $3$-$11$, "... outside of some closed set contained in $U$", the word "closed" should be replaced with "compact". So, property (4) can be rephrased equivalently by requiring that the support of $\varphi$ be a compact subset of $U$, where the support is defined as the topological closure of the set of points where $\varphi$ is non-zero. \begin{equation} \text{supp}(\varphi) := \overline{\{ x \in \mathbb{R^n}: \varphi(x)\neq 0\}}. \end{equation}
Next, to define the extended integral, I think this is a better definition (it's almost the same, but there are a few subtle differences):
The two differences are: I only required $\Phi$ to be $\mathcal{C^0}$, not $\mathcal{C^{\infty}}$, and second, I put $\displaystyle \int_{\text{supp}(\varphi)} \varphi \cdot |f|$ rather than $\displaystyle \int_{A} \varphi \cdot |f|$. The reason I made the second change is because the purpose of this definition is to define integration on an open set (which may be unbounded), so writing $\displaystyle \int_{A} \varphi \cdot |f|$ isn't even defined based on all the old definitions. However, this isn't a huge deal, because later on we can show that \begin{equation} (\text{extended})\displaystyle \int_{A} \varphi \cdot |f| = (\text{old}) \displaystyle \int_{\text{supp}(\varphi)} \varphi \cdot |f| \end{equation} But, from a logical standpoint, we should not use the symbol $\displaystyle \int_A \varphi \cdot f$ in a definition where we're trying to define the meaning of integration on $A$ (note that we have to use another partition of unity $\Psi$ to make sense of the LHS above).
Remarks: