Riemann Integration in \({\mathbb R}^n\)
1. Integration on Boxes
If \(R \subset {\mathbb R}^n\) is a box and \(f\) is a bounded real-valued function on \(R\), then
\[ \begin{aligned} {\mathcal U}(f,{\mathcal P}) & := \sum_{R' \in \mathcal P} |R'| \sup_{x \in R'} f(x), \\ {\mathcal L}(f,{\mathcal P}) & := \sum_{R' \in \mathcal P} |R'| \inf_{x \in R'} f(x). \end{aligned}\]
Some facts whose proofs are very similar to 1D theory:
- \({\mathcal L}(f,{\mathcal P}) \leq {\mathcal U}(f,{\mathcal P})\)
- After refining a partition, upper sums are never larger and lower sums are never smaller.
- Any upper sum on any partition dominates any lower sum on any other partition.
- For any bounded functions \(f\) and \(g\),\[{}{\mathcal U}(f+g,{\mathcal P}){}\]\[{}\leq {\mathcal U}(f,{\mathcal P}) + {\mathcal U}(g,{\mathcal P}){}\]\[{}{\mathcal L}(f+g,{\mathcal P}),{}\]\[{}\geq {\mathcal L}(f,{\mathcal P}) + {\mathcal L}(g,{\mathcal P}).{}\]
- For positive scalar \(c\),\[{}{\mathcal U} (cf,{\mathcal P}){}\]\[{}= c {\mathcal U}(f,{\mathcal P}),{}\]\[{}{\mathcal L} (cf,{\mathcal P}){}\]\[{}= c {\mathcal L}(f,{\mathcal P}).{}\]
The function \(f\) is called Riemann integrable on \(R\) when the infimum of upper sums equals the supremum of lower sums; define this number to be the integral:
\[ \int_R f ~dx := \inf_{{\mathcal P}} {\mathcal U}(f,{\mathcal P}) = \sup_{{\mathcal P}} {\mathcal L}(f,{\mathcal P}). \]
Proposition (Integration Facts)
- If \(f,g\) are Riemann integrable on \(R\), then so is \(c f + g\), where \(c\) is a constant, and\[ \int_R (c f + g) dx = c \int_R f ~dx + \int_R g ~dx. \]
- If \(f\) is Riemann integrable and \(\phi : I \rightarrow {\mathbb R}\) is Lipschitz on some interval \(I\) containing the range of \(f\) (recall: \(|\phi(t_1) - \phi(t_2)|\) \(\leq C |t_1 - t_2|\)), then \(\phi \circ f\) is Riemann integrable.
Proof
2. Integration on Jordan Regions
Corollary (Squaring Preserves Riemann Integration)
If \(f\) is integrable on \(R\), then \(f^2\) is too.
Corollary (Products of Integrable Functions are Integrable)
If \(f,g\) are Riemann integrable on \(R\), then \(fg\) is too.
Surprise! A set \(E\) is a Jordan region if and only if its characteristic function \(\chi_E\) is Riemann integrable.
We say that a bounded function \(f\) on a Jordan region is Riemann integrable on \(E\) if \(f \chi_E\) is Riemann integrable and we define
\[ \int_E f ~dx := \int f \chi_E ~dx. \]
3. Comparison Theorems
Theorem
- If \(f \leq g\) are both integrable on Jordan region \(R\), then \(\int_R f ~ dx \leq \int_R g ~ dx\).
- If \(f\) is integrable on \(R\) then \(|f|\) is too and\[\left| \int_R f dx\right| \leq \int_R |f| dx \](This is called the triangle inequality or Minkowski inequality).
- If \(m \leq f \leq M\), then\[m \text{Vol}(R) \leq \int_R f ~ dx \leq M \text{Vol}(R).\]
Exercises
- Suppose that \(f\) is a nonnegative Riemann integrable function on a closed box \(R \subset {\mathbb R}^d\). Show that \(\int_R f(x) dx = 0\) implies that \(f(x) = 0\) at some point in \(R\).HintFix any \(\epsilon > 0\) and show that there is some partition \(\mathcal P\) of \(R\) such that \(\sup_{x \in R'} f(x) < \epsilon\) for some \(R' \in \mathcal P\). Then use the identity \(f \chi_{R'} \leq f\) to conclude that \(\int_{R'} f(x) dx = 0\) as well. Conclude that there exists a nested sequence of closed boxes \(R'_1 \supset R'_2 \supset \cdots\) such that \(\sup_{x \in R'_j} f(x) \rightarrow 0\) as \(j \rightarrow \infty\). Conclude that there is a point \(x\) in the intersection of these boxes at which \(f(x) = 0\).
- Generalize the previous result to the following: if \(E\) is a Jordan region, \(f\) is Riemann integrable on \(E\) with \(f(x) > c\) for each \(x \in {\mathbb R}\), and \(w\) is Riemann integrable on \(E\) with \(w(x) \geq 0\) and \(\int_E w(x) dx > 0\), then\[ \int_E f(x) w(x) dx > c \int_E w(x) dx. \]HintAssume that \(\int_E (f(x)-c) w(x) dx = 0\). show that there must be some closed box \(R \subset E\) on which \(w\) is bounded uniformly below by a positive constant (think \(\mathcal{L}(w \chi_E,\mathcal P)\)) and prove that there is a point in this box at which \(f(x) = c\).