Advanced Analysis

Riemann Integration in \({\mathbb R}^n\)

Video. Riemann Integration

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
Meta (Key Point)
For \(x,y \in R'\),
\[{}\phi(f(x)) - \phi(f(y)){}\]
\[{}\leq C \max\{ f(x) - f(y), f(y) - f(x) \}{}\]
\[{}\leq C (\sup_{u \in R'} f(u) - \inf_{v \in R'} f(v)){}\]

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
\[ \int_E f ~dx := \int f \chi_E ~dx. \]
Proposition (Integrability of Continuous Functions)
Continuous functions are integrable on Jordan regions.
Proof
Meta (Main Idea)
Continuous functions are always integrable on boxes because they are uniformly continuous there (because boxes are compact). Because characteristic functions of Jordan regions are also Riemann integrable, continuous functions are always integrable on Jordan regions.

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
  1. 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 \(R\).
    Hint
    Fix 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 \(f(x) = 0\).
  2. 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
    \[ \int_E f(x) w(x) dx > c \int_E w(x) dx. \]
    Hint
    Assume 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 \(f(x) = c\).