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Math412 Notes Select Topics in Mathematical Analysis of Functions of One or Several Variables

Zheng-Chao Han

Section 6.1 Basic Definitions and Properties for the Integral of a Scalar Function on a Multi-dimensional Rectangle

There is little difference between one and higher dimensions in the discussion of this and next section; differences will show when discussing evaluation of integrals and change of variables.
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Definition 6.1.1.

A set of the form \(R=[a_{1}, b_{1}]\times \cdots \times [a_{n}, b_{n}]\) , where \(a_{i} < b_{i}\) for each \(i\text{,}\) is called a hypercube in \(\bbR^{n}\text{.}\) We also informally call it an \(n\)-dimensional rectangle (or rectangular box). We define its volume to be
\begin{equation*} |R| := (b_{1}-a_{1})\cdots (b_{n}-a_{n}). \end{equation*}
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When \(a_{i}=b_{i}\) is allowed, and this equality holds for at least one \(i\text{,}\) we call such an \(R\) a degenerate hypercube (or informally a degenerate rectangle), and define its volume to be \(0\text{.}\) A hypercube \(R\) has faces, edges and vertices obtained when one or more of the \(x_{i}\) variables is held as fixed at either \(a_{i}\) or \(b_{i}\text{,}\) which are degenerate hypercubes.
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Suppose that \(\cP_{i}\) is a partition of \([a_{i}, b_{i}]\) for \(1\le i \le n\text{.}\) Then these partitions \(\cP_{i}\)’s form a partition \(\cP\) of \(R=[a_{1}, b_{1}]\times \cdots \times [a_{n}, b_{n}]\) in the following sense:
  1. The cells of the partition \(\cP\) consist of
    \begin{equation*} \{S:=I_{1}\times\cdots \times I_{n}: I_{i} \text{ is a subinterval of } \cP_{i}\}\text{;} \end{equation*}
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  2. The union of the cells of \(\cP =R\text{;}\)
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  3. The intersection of any two different cells of \(\cP\) is either the empty set, or a degenerate rectangle; and
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  4. \(\vert R\vert=\sum_{S\in \cP}\vert S\vert\text{.}\)
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We may denote this partition by \(\cP_{1}\times\cdots \times \cP_{n}\text{.}\)
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The partition size \(\lambda(\cP)\) of \(\cP:=\cP_{1}\times\cdots \times \cP_{n}\) is defined as
\begin{equation*} \lambda(\cP) :=\max\{\vert I_{i}\vert: S:=I_{1}\times\cdots\times I_{n} \in \cP\}. \end{equation*}
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Remark 6.1.2.

Only the verification of the third and last property requires more than a few lines of argument. To prove (iii), note that a partition of \(R\) in our definition is defined in terms of a collection of partitions \(\cP_{i}\) for each constitutive factor. As a result, if the intersection of two rectangles \(S_{1}, S_{2}\) in a partition of \(R\) is non-empty, then for each \(i\text{,}\) the constitutive factor \(I_{i, 1}\) for \(S_{1}\) and respectively \(I_{i, 2}\) for \(S_{2}\) must have non-empty intersection. This means that \(I_{i, 1}\) and \(I_{i, 2}\) either share one common end point or are identical. We can now conclude that if two rectangles \(S_{1}, S_{2}\) in a partition of \(R\) have non-empty intersection, then the intersection must be a degenerate hypercube.
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For (iv), one easy way is to use an induction argument on \(n\text{,}\) the dimension. \(R':=[a_{2}, b_{2}]\times \cdots \times [a_{n}, b_{n}]\) is an \((n-1)\)-dimensional rectangle, and the partitions \(\{\cP_{2},\ldots, \cP_{n}\}\) form a partition \(\cP'\) of \(R'\text{.}\) Suppose that \(\{I_{1,1},\ldots, I_{1,k}\}\) are the subintervals of \(\cP_{1}\text{.}\) In the summation \(\sum_{S\in \cP}\vert S\vert\text{,}\) group \(S\) according to its constitutive factor in the first variable: each \(S\) has the form of \(I_{1,j}\times S'\) for some \(I_{1,j}\in \cP_{1}\) and \(S'\in \cP'\text{,}\) thus we have
\begin{equation*} \sum_{S\in \cP}\vert S\vert=\sum_{I_{1,j}\in \cP_{1}}\sum_{S'\in \cP'} \vert I_{1,j} \times S'\vert. \end{equation*}
By induction, we have
\begin{equation*} \sum_{S'\in \cP'} \vert I_{1,j} \times S'\vert= \vert I_{1,j}\vert \sum_{S'\in \cP'} \vert S'\vert= \vert I_{1,j}\vert \vert R'\vert. \end{equation*}
It now follows that
\begin{equation*} \sum_{S\in \cP}\vert S\vert=\sum_{I_{1,j}\in \cP_{1}} \vert I_{1,j}\vert \vert R'\vert =(b_{1}-a_{1}) \vert R'\vert=\vert R\vert. \end{equation*}
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One could allow more general choices of rectangles in a partition; it would just make it harder to bookkeep such rectangles, and our definition suffices for our purpose of defining the integral of a function.
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Definition 6.1.3.

Suppose that \(f\) is a bounded real-valued function defined on the rectangle \(R\text{,}\) \(\cP\) a partition of \(R\text{,}\) and a point \(\bx_{\alpha}\in S_{\alpha}\) is chosen for each sub rectangle \(S_{\alpha}\) of \(\cP\text{.}\) We define the corresponding Riemann sum as
\begin{equation*} R(f, \cP, \{\bx_{\alpha}\}):=\sum_{\alpha} f(\bx_{\alpha}) \vert S_{\alpha} \vert. \end{equation*}
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Define
\begin{equation*} M_{S}(f):=\sup_{\bx\in S}f(\bx) \quad \text{and} \quad m_{S}(f):=\inf_{\bx\in S}f(\bx) \end{equation*}
as the supremum, and respectively infimum, of \(f\) on the rectangle \(S\text{.}\) Then the upper sum \(U(f, \cP)\text{,}\) and respectively the lower sum \(L(f, \cP)\text{,}\) of \(f\) on \(R\) with respect to the partition \(\cP\) is defined as
\begin{equation*} U(f, \cP) :=\sum_{\alpha} M_{S_{\alpha}}(f) \vert S_{\alpha}\vert, \quad L(f, \cP) :=\sum_{\alpha} m_{S_{\alpha}}(f) \vert S_{\alpha}\vert. \end{equation*}
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Note that
\begin{equation*} L(f, \cP)\le R(f, \cP, \{\bx_{\alpha}\})\le U(f, \cP). \end{equation*}
We are interested in whether \(R(f, \cP, \{\bx_{\alpha}\})\) has a limit as \(\lambda (\cP)\to 0\) which is independent of how \(\bx_{\alpha}\in S_{\alpha}\) is chosen. It is easier to study whether \(U(f, \cP)\text{,}\) and respectively \(L(f, \cP)\text{,}\) has a limit as \(\lambda (\cP)\to 0\text{.}\)
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Definition 6.1.4.

A partition \(\cP^{*}:=\cP^{*}_{1}\times\cdots \times \cP^{*}_{n}\) is called a refinement of partition \(\cP:=\cP_{1}\times\cdots \times \cP_{n}\) if each \(\cP^{*}_{i}\) is a refinement of \(\cP_{i}\text{.}\)
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Figure 6.1.5. An illustration of a common refinement of two given partitions generated with the help of Gemini
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Note that if \(\cP^{*}\) is a refinement of partition \(\cP\text{,}\) then \(\lambda(\cP^{*})\le \lambda(\cP)\text{,}\) and that
\begin{equation*} U(f, \cP^{*})\le U(f, \cP), L(f, \cP^{*})\ge L(f, \cP). \end{equation*}
Just as in one dimension, if \(\cP_{1}, \cP_{2}\) are two partitions of \(R\text{,}\) then there is a partition \(\cP^{*}\) which is a refinement of both \(\cP_{1}\) and \(\cP_{2}\text{,}\) and there is an essentially canonical way of constructing such an \(\cP^{*}\) by adjoining all the end points of the subintervals of the factors of \(\cP_{1}, \cP_{2}\text{.}\)
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Definition 6.1.6.

Suppose that \(f\) is a bounded function defined on the rectangle \(R\subset \bbR^{n}\text{.}\) Then its upper integral, and respectively lower integral, on \(R\) is defined as \(\inf_{\cP}U(f, \cP)\text{,}\) and respectively \(\sup_{\cP}L(f, \cP)\text{,}\) where \(\cP\) runs over all partitions of \(\cP\text{.}\) The upper integral is denoted as \(\upint_{R}\,f\text{,}\) while the lower integral is denoted as \(\lowint_{\;R}\,f\text{.}\)
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Exercise 6.1.7.

Let \(\cC\) denote the standard tertiary Cantor set on \([0, 1]\) and \(\chi_{\cC}\) denote its characteristic function which takes value \(1\) on \(\cC\) and \(0\) elsewhere. Let \(\cP\) be a partition of \([0, 1]\) into intervals of equal length \(3^{-k}\) for some \(k\in \bbN\text{.}\) Find \(U(\chi_{\cC}, \cP)\) and \(L(\chi_{\cC}, \cP)\text{.}\) Is there a positive lower bound of \(L(\chi_{\cC}, \cP)\) independent of \(\cP\text{?}\)
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Proof.

Let \(\cP_{1}, \cP_{2}\) be two arbitrary partitions of \(R\text{,}\) and \(\cP^{*}\) be a refinement of both \(\cP_{1}\) and \(\cP_{2}\text{.}\) Then
\begin{equation*} L(f, \cP_{1})\le L(f, \cP^{*})\le U(f, \cP^{*})\le U(f, \cP_{2}). \end{equation*}
As a result,
\begin{equation*} L(f, \cP_{1})\le \upint_{R} \, f =\inf_{\cP_{2}} U(f, \cP_{2}), \end{equation*}
and
\begin{equation*} \lowint_{\;R} \, f =\sup_{\cP_{1}} L(f, \cP_{1})\le \upint_{R} \, f. \end{equation*}
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For any \(\epsilon>0\text{,}\) first find a partition \(\cP_{2}\) of \(R\) such that
\begin{equation*} \upint_{R} \, f \le U(f, \cP_{2}) \le \upint_{R} \, f + \epsilon. \end{equation*}
Now take any partition \(\cP\) of \(R\) of small partition size \(\lambda(\cP)\) (to be determined). Let \(\cP^{*}\) be the canonical refinement of \(\cP\) and \(\cP_{2}\text{.}\) Then
\begin{equation*} \upint_{R} \, f \le U(f, \cP_{*}) \le U(f, \cP_{2}) \le \upint_{R} \, f + \epsilon. \end{equation*}
and there exists a number \(N\) depending only on the partition \(\cP_{2}\) such that in the sums \(U(f,\cP)\) and \(U(f, \cP^{*})\text{,}\) all but at most \(N\) terms may differ, as in the construction of \(\cP^{*}\text{,}\) only those rectangles in the partition of \(\cP\) whose projection into some \(x_{i}\) variable contains a partition point corresponding to the projection of \(\cP_{2}\) need to be refined, and the number of such sub intervals of the factors of \(\cP\) that need to be refined in relation to \(\cP_{2}\) has a bound that depends only on \(\cP_{2}\text{.}\)
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Note that if \(S\) is such a rectangle, then after refinement in relation to \(\cP_{2}\text{,}\) \(S\) is the non-overlapping union \(\cup_{k=1}^{l}S_{k}\text{,}\) so \(|S|=\sum_{k=1}^{l} |S_{k}|\text{,}\) and
\begin{equation*} 0\le M_{S}(f)|S|- \sum_{k=1}^{l}M_{S_{k}}(f)|S_{k}|\le 2\max_{R}|f| |S| \end{equation*}
with \(|S|\) bounded above by \((\lambda(\cP))^{n}\text{.}\) Thus
\begin{equation*} \vert U(f,\cP)- U(f, \cP^{*})\vert \le 2 N \max_{R}|f| (\lambda(\cP))^{n}. \end{equation*}
It now follows that there exists some \(\delta >0\) such that whenever \(\lambda(\cP) < \delta\) we have
\begin{equation*} \vert U(f,\cP)- U(f, \cP^{*})\vert < \epsilon \text{ so } U(f,\cP) < \upint_{R} \, f + 2\epsilon, \end{equation*}
proving that \(\lim_{\lambda(\cP)\to 0} U(f, \cP)= \upint_{R} \, f\text{.}\)
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The proof of \(\sup_{\cP}L(f, \cP) = \lim_{\lambda(\cP)\to 0} L(f, \cP)\) is done in a similar way.
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Exercise 6.1.9.

Suppose that \(f, g\) are bounded functions on the rectangle \(R\text{.}\) Show that
\begin{equation*} L(f,\cP)+L(g, \cP)\le L(f+g, \cP) \text{ and } U(f+g, \cP)\le U(f, \cP) +U(g, \cP)\text{.} \end{equation*}
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Definition 6.1.10.

A bounded real-valued function \(f\) defined on a rectangle \(R\) is called Riemann integrable, if there exists a real number \(S\) such that \(\lim_{\lambda(P)\to 0} R(f, \cP, \{\bx_{\alpha}\})=S\) in the sense that for any \(\epsilon >0\text{,}\) there exists some \(\delta >0\) that
\begin{equation*} \vert R(f, \cP, \{\bx_{\alpha}\}) -S\vert < \epsilon \end{equation*}
for any partition \(\cP\) and choice of \(\bx_{\alpha}\in R_{\alpha}\in \cP\text{,}\) as long as \(\lambda(P) < \delta\text{.}\)
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When such an \(S\) exists, it is unique, and we denote it by \(\int_{R}\, f\text{.}\)
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Proof.

Let \(S_{1}= \int_{R} f_{1}\) and \(S_{2}= \int_{R} f_{2}\text{.}\) Then for any \(\epsilon >0\text{,}\) there exists \(\delta>0\) such that for any partition \(\cP\) of \(R\text{,}\) whenever \(\lambda (\cP) < \delta\text{,}\) we have
\begin{gather*} \vert R(f_{1}, \cP, \{\bx_{\alpha}\}) -S_{1}\vert < \epsilon,\\ \vert R(f_{2}, \cP, \{\bx_{\alpha}\}) -S_{2}\vert < \epsilon. \end{gather*}
Then
\begin{align*} \amp \vert R(c_{1}f_{1}+c_{2}f_{2}, \cP, \{\bx_{\alpha}\}) - \left(c_{1}S_{1}+c_{2 }S_{2} \right)\vert \\ \le \amp |c_{1}| \vert R(f_{1}, \cP, \{\bx_{\alpha}\}) -S_{1}\vert + |c_{2}| \vert R(f_{2}, \cP, \{\bx_{\alpha}\}) -S_{2}\vert \vert \\ < \amp \left(|c_{1}|+|c_{2}| \right) \epsilon, \end{align*}
which shows the integrability of \(c_{1}f_{1}+c_{2}f_{2}\) as well as (6.1.1).
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Based on Proposition 6.1.8, a necessary condition that \(f \) be Riemann integrable on \(R\) is that
\begin{equation} \lowint_{\;R} \, f =\upint_{R}\, f.\tag{6.1.2} \end{equation}
This turns out to be also sufficient.
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Proof.

We only need to prove the if part. Suppose (6.1.2) holds. Call the value on both sides \(\int_{R}\, f\text{.}\) Our proof in Proposition 6.1.8 essentially carries over to show that, for any \(\epsilon >0\text{,}\) there exists some \(\delta >0\text{,}\) such that for any partition \(\cP\) of \(R\) with \(\lambda(\cP) < \delta\text{,}\) we have
\begin{equation*} \int_{R}\, f-\epsilon < L(f, \cP) \le \int_{R}\, f \le U(f, \cP) < \int_{R}\, f +\epsilon. \end{equation*}
Then for any choice of \(\bx_{\alpha}\in R_{\alpha}\in \cP\text{,}\) we have
\begin{equation*} \int_{R}\, f-\epsilon < L(f, \cP) \le R(f, \cP, \{\bx_{\alpha}\})\le U(f, \cP) < \int_{R}\, f +\epsilon, \end{equation*}
which shows that \(f\) is Riemann integrable on \(R\text{.}\)
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Example 6.1.13.

Let
\begin{equation*} f(x)=\begin{cases} 1 \quad x \in [0, 1] \text{ rational, } \\ 0 \quad x \in [0, 1] \text{ irrational; } \end{cases} \end{equation*}
\begin{equation*} g(x, y) =\begin{cases} 1 \quad x \in [0, 1] \text{ rational, } y=0\\ 0 \quad \text{ elsewhere in } [0, 1]\times[0, 1]; \end{cases} \end{equation*}
and \(h(x, y)=f(x)f(y)\text{.}\)
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\(f\) is discontinuous everywhere on \([0,1]\text{,}\) \(g\) is continuous on \([0,1]\times (0, 1]\text{,}\) but is discontinuous on \([0, 1]\times \{0\}\text{,}\) while \(h\) is discontinuous everywhere on \([0,1]\times[0,1]\text{.}\)
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For any partition \(\cP_{1}: 0=s_{0} < s_{1} < \cdots < s_{k}=1 \) of \([0, 1]\) in the \(x\) variable, \(U(f, \cP_{1})=1\) and \(L(f, \cP_{1})=0\text{.}\) Thus \(\lowint_{\;[0,1]}\, f(x)\, dx =0\) , \(\upint_{[0, 1]}\, f(x)\, dx =1\text{,}\) and \(f\) is not Riemann integrable on \([0, 1]\text{.}\)
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For any partition \(\cP_{2}: 0=t_{0} < t_{1} < \cdots < t_{l}=1 \) of \([0, 1]\) in the \(y\) variable, \(m(g, [s_{i-1},s_{i}]\times[t_{j-1}, t_{j} ])=0\) for all \(i, j\text{,}\) and
\begin{equation*} M(g, [s_{i-1},s_{i}]\times[t_{j-1}, t_{j} ])=\begin{cases} 1 \quad j=0,\\ 0 \quad j>1. \end{cases} \end{equation*}
Thus \(L(g, \cP_{1}\times \cP_{2})=0\text{,}\) \(U(g, \cP_{1}\times \cP_{2})=t_{1}\text{,}\) and \(\lowint_{\;[0,1]\times [0,1]}\, g =0\text{,}\) \(\upint_{[0,1]\times [0,1]}\, g =0\text{,}\) so we conclude that \(g\) is Riemann integrable on \([0,1]\times [0,1]\text{,}\) and \(\int_{[0,1]\times [0,1]} g =0\text{.}\)
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For \(h\text{,}\) we have \(m(h, [s_{i-1},s_{i}]\times[t_{j-1}, t_{j} ])=0, m(h, [s_{i-1},s_{i}]\times[t_{j-1}, t_{j} ])=1\) for all \(i, j\text{,}\) so \(L(h, \cP_{1}\times \cP_{2})=0\text{,}\) \(U(g, \cP_{1}\times \cP_{2})=1\text{.}\) As a result, \(\lowint_{\;[0,1]\times [0,1]}\, h =0\text{,}\) \(\upint_{[0,1]\times [0,1]}\, h =1\text{,}\) so \(h\) is not Riemann integrable on \([0,1]\times [0,1]\text{.}\)
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Exercise 6.1.14.

Let \(0 \lt r_{k} \lt 1\) be such that \(\sum_{k=1}^{\infty} r_{k} \lt \infty\text{.}\) Define a Cantor set \(\cK\) on \([0, 1]\) by removing the middle \(r_{k}\) portion of the remaining intervals at stage \(k\text{.}\) Let \(\chi_{\cK}\) denote its characteristic function. Is \(\chi_{\cK}\) Riemann integrable on \([0, 1]\text{?}\)
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