5.7 Integration in the plane  (Page 4/5)

Of course, we could now extend the notion of integrability over a geometric set $S$ to include complex-valued functions just as we did for integrability over an interval $[a,b].$ However, real-valued functions on geometric sets will suffice for the purposes of this book.

We include here, to be used later in [link] , a somewhat technical theorem about constructing partitions of a geometric set.

Let $S_1,...,S_n$ be closed, nonoverlapping, geometric sets, all contained in a geometric set $S.$ Then there exists a partition ${\widehat{S}}_1,...,{\widehat{S}}_M$ of $S$ such that for $1\le i\le n$ we have $S_i={\widehat{S}}_i.$ In other words, the $s_i$ 's are the first $n$ elements of a partition of $S.$

Suppose $S$ is determined by the interval $[a,b]$ and the two bounding functions $u$ and $l.$ We prove this theorem by induction on $n.$

If $n=1,$ let $S_1$ be determined by the interval $[a_1,b_1]$ and the two bounding functions $u_1$ and $l_1.$ Set ${\widehat{S}}_1=S_1,$ and define four more geometric sets ${\widehat{S}}_2,...,{\widehat{S}}_5$ as follows:

1.   ${\widehat{S}}_2$ is determined by the interval $[a,a_1]$ and the two bounding functions $u$ and $l$ restricted to that interval.
2.   $S_3$ is determined by the interval $[a_1,b_1]$ and the two bounding functions $u$ and $u_1$ restricted to that interval.
3.   $S_4$ is determined by the interval $[a_1,b_1]$ and the two bounding functions $l$ and $l_1$ restricted to that interval.
4.   ${\widehat{S}}_5$ is determined by the interval $[b_1,b]$ and the two bounding functions $u$ and $l$ restricted to that interval.

Observe that the five sets ${\widehat{S}}_1,{\widehat{S}}_2,...,{\widehat{S}}_5$ constitute a partition of the geometric set $S,$ proving the theorem in the case $n=1.$

Suppose next that the theorem is true for any collection of $n$ sets satisfying the hypotheses. Then, given $S_1,...,S_{n+1}$ as in the hypothesis of the theorem, apply the inductive hypothesis to the $n$ sets $S_1,...,S_n$ to obtain a partition $T_1,...,T_m$ of $S$ for which $T_i=S_i$ for all $1\le i\le n.$ For each $n+1\le i\le m,$ consider the geometric set $S_i^{\text{'}}=S_{n+1}\cap T_i$ of the geometric set $T_i.$ We may apply the case $n=1$ of this theorem to this geometric set to conclude that $S_i^{\text{'}}$ is the first element $S_{i,1}^{\text{'}}$ of a partition $\{S_{i,1}^{\text{'}},S_{i,2}^{\text{'}},...,S_{i,m_i}^{\text{'}}\}$ of the geometric set $T_i.$

Define a partition $\left\{{\widehat{S}}_k\right\}$ of $S$ as follows: For $1\le k\le n,$ set ${\widehat{S}}_k=T_k.$ Set ${\widehat{S}}_{n+1}={\cup }_{i=n+1}^m{S}_{i,1}^{\text{'}}=S_{n+1}.$ And define the rest of the partition $\left\{{\widehat{S}}_k\right\}$ to be made up of the remaining sets $S_{i,j}^{\text{'}}$ for $n+1\le i\le m$ and $2\le j\le m_i.$ It follows directly that this partition $\left\{{\widehat{S}}_k\right\}$ satisfies the requirements of the theorem.