5.7 Integration in the plane  (Page 3/5)

Let $H\left(S\right)$ denote the vector space of all step functions on the closed geometric set $S.$ Then the assignment $h\to \int h$ of $H\left(S\right)$ into $R$ has the following properties:

1.  (Linearity)  $H\left(S\right)$ is a vector space, and ${\int }_S(h_1+h_2)={\int }_S{h}_1+{\int }_S{h}_2,$ and ${\int }_{S}ch=c{\int }_{S}h$ for all $h_1,h_2,h\in H\left(S\right),$ and for all real numbers $c.$
2. If $h={\sum }_{i=1}^n{c}_i{\chi }_{S_i}$ is a linear combination of indicator functions of geometric sets that are subsets of $S,$ then $\int h={\sum }_{i=1}^n{c}_{i}A\left(S_i\right).$
3.  (Positivity) If $h\left(z\right)\ge 0$ for all $z\in S,$ then ${\int }_{S}h\ge 0.$
4.  (Order-preserving) If $h_1$ and $h_2$ are step functions on $S$ for which $h_1\left(z\right)\le h_2\left(z\right)$ for all $z\in S,$ then ${\int }_S{h}_1\le {\int }_S{h}_2.$

Suppose $h_1$ is constant on the elements of a partition $P=\left\{S_i\right\}$ and $h_2$ is constant on the elements of a partition $Q=\left\{T_j\right\}.$ Let $V$ be the partition of the geometric set $S$ whose elements are the sets $\left\{U_k\right\}=\{S_i^0\cap T_j^0\}.$ Then both $h_1$ and $h_2$ are constant on the elements $U_k$ of $V,$ so that $h_1+h_2$ is also constant on these elements. Therefore, $h_1+h_2$ is a step function, and

and this proves the first assertion of part (1).

The proof of the other half of part (1), as well as parts (2), (3), and (4), are totally analogous to the proofs of the corresponding parts of [link] , and we omit the arguments here.

Now for the other necessary consistency condition:

let $S$ be a closed geometric set in the plane.

1. If $\left\{h_n\right\}$ is a sequence of step functions that converges uniformly to a function $f$ on $S,$ then the sequence $\left\{{\int }_S{h}_n\right\}$ is a convergent sequence of real numbers.
2. If $\left\{h_n\right\}$ and $\left\{k_n\right\}$ are two sequences of step functions on $S$ that converge uniformly to the same function $f,$ then
   $$lim{\int }_S{h}_n=lim{\int }_S{k}_n.$$

If $f$ is a real-valued function on a closed geometric set $S$ in the plane, then $f$ is integrable on S if it is the uniform limit of a sequence $\left\{h_n\right\}$ of step functions on $S.$

We define the integral of an integrable function $f$ on $S$ by

$${\int }_{S}f\equiv {\int }_{S}f\left(z\right)dz=lim{\int }_S{h}_n,$$

where $\left\{h_n\right\}$ is a sequence of step functions on $S$ that converges uniformly to $f.$

Let $S$ be a closed geometric set in the plane, and let $I\left(S\right)$ denote the set of integrable functions on $S.$ Then:

1.   $I\left(S\right)$ is a vector space of functions.
2. If $f$ and $g\in I\left(S\right),$ and one of them is bounded, then $fg\in I\left(S\right).$
3. Every step function is in $I\left(S\right).$
4. If $f$ is a continuous real-valued function on $S,$ then $f$ is in $I\left(S\right).$ That is, every continuous real-valued function on $S$ is integrable on $S.$

Let $S$ be a closed geometric set. The assignment $f\to \int f$ on $I\left(S\right)$ satisfies the following properties.

1.  (Linearity)  $I\left(S\right)$ is a vector space, and ${\int }_S(\alpha f+\beta g)=\alpha {\int }_{S}f+\beta {\int }_{S}g$ for all $f,g\in I\left(S\right)$ and $\alpha ,\beta \in R.$
2.  (Positivity) If $f\left(z\right)\ge 0$ for all $z\in S,$ then ${\int }_{S}f\ge 0.$
3.  (Order-preserving) If $f,g\in I\left(S\right)$ and $f\left(z\right)\le g\left(z\right)$ for all $z\in S,$ then ${\int }_{S}f\le {\int }_{S}g.$
4. If $f\in I\left(S\right),$ then so is $\left|f\right|,$ and $|{\int }_{S}f|\le {\int }_S\left|f\right|.$
5. If $f$ is the uniform limit of functions $f_n,$ each of which is in $I\left(S\right),$ then $f\in I\left(S\right)$ and ${\int }_{S}f=lim{\int }_S{f}_n.$
6. Let $\left\{u_n\right\}$ be a sequence of functions in $I\left(S\right),$ and suppose that for each $n$ there is a number $m_n,$ for which $|u_n\left(z\right)|\le m_n$ for all $z\in S,$ and such that the infinite series $\sum m_n$ converges. Then the infinite series $\sum u_n$ converges uniformly to an integrable function, and ${\int }_S\sum u_n=\sum {\int }_S{u}_n.$
7. If $f\in I\left(S\right),$ and $\{S_1,...,S_n\}$ is a partition of $S,$ then $f\in I\left(S_i\right)$ for all $i,$ and
   $${\int }_S=\sum _{i=1}^n{\int }_{S_i}f.$$