5.2 Integrable functions  (Page 2/4)

Define a function $f$ on the closed interval $[0,1]$ by $f\left(x\right)=x.$

1. For each positive integer $n,$ let $P_n$ be the partition of $[0,1]$ given by the points $\{0<1/n<2/n<3/n<...<(n-1)/n<1\}.$ Define a step function $h_n$ on $[0,1]$ by setting $h_n\left(x\right)=i/n$ if $\frac{i-1}{n}<x<\frac{i}{n},$ and $h_n(i/n)=i/n$ for all $0\le i\le n.$ Prove that $|f\left(x\right)-h_n\left(x\right)|<1/n$ for all $x\in [0,1],$ and then conclude that $f$ is the uniform limit of the $h_n$ 's whence $f\in I\left(\right[0,1\left]\right).$
2. Show that
   $$\int h_n=\sum _{i=1}^n\frac{i}{n^2}=\frac{n(n+1)}{2n^2}.$$
3. Show that ${\int }_0^{1}f\left(t\right)dt=1/2.$ The next exercise establishes some additional properties of integrable functions on an interval $[a,b].$

Let $[a,b]$ be a closed and bounded interval, and let $f$ be an element of $I\left(\right[a,b\left]\right).$

1. Show that, for each $ϵ>0$ there exists a step function $h$ on $[a,b]$ such that $\left|f\right(x)-h(x\left)\right|<ϵ$ for all $x\in [a,b].$
2. For each positive integer $n$ let $h_n$ be a step function satisfying the conclusion of part (a) for $ϵ=1/n.$ Define $k_n=h_n-1/n$ and $l_n=h_n+1/n.$ Show that $k_n$ and $l_n$ are step functions, that $k_n\left(x\right)<f\left(x\right)<l_n\left(x\right)$ for all $x\in [a,b],$ and that $|l_n\left(x\right)-k_n\left(x\right)|=l_n\left(x\right)-k_n\left(x\right)=2/n$ for all $x.$ Hence, ${\int }_a^b(l_n-k_n)=\frac{2}{n}(b-a).$
3. Conclude from part (b) that, given any $ϵ>0,$ there exist step functions $k$ and $l$ such that $k\left(x\right)\le f\left(x\right)\le l\left(x\right)$ for which $\int \left(l\right(x)-k(x\left)\right)<ϵ.$
4. Prove that there exists a sequence $\left\{j_n\right\}$ of step functions on $[a,b],$ for which $j_n\left(x\right)\le j_{n+1}\left(x\right)\le f\left(x\right)$ for all $x,$ that converges uniformly to $f.$ Show also that there exists a sequence $\left\{j_n^{\text{'}}\right\}$ of step functions on $[a,b],$ for which $j_n^{\text{'}}\left(x\right)\ge j_{n+1}^{\text{'}}\left(x\right)\ge f\left(x\right)$ for all $x,$ that converges uniformly to $f.$ That is, if $f\in I\left(\right[a,b\left]\right),$ then $f$ is the uniform limit of a nondecreasing sequence of step functions and also is the uniformlimit of a nonincreasing sequence of step functions. HINT: To construct the $j_n$ 's and $j_n^{\text{'}}$ 's, use the step functions $k_n$ and $l_n$ of part (b), and recall that the maximum and minimum of step functions is again a step function.
5. Show that if $f\left(x\right)\ge 0$ for all $x\in [a,b],$ and $g$ is defined by $g\left(x\right)=\sqrt{f\left(x\right)},$ then $g\in I\left(\right[a,b\left]\right).$ HINT: Write $f=lim{h}_n$ where $h_n\left(x\right)\ge 0$ for all $x$ and $n.$ Then use part (g) of Exercise 3.28.
6. (Riemann sums again.) Show that, given an $ϵ>0,$ there exists a partition $P$ such that if $Q=\{x_0<x_1<...<x_n\}$ is any partition finer than $P,$ and $\left\{w_i\right\}$ are any points for which $w_i\in (x_{i-1},x_i),$ then
   $$|{\int }_a^{b}f\left(t\right)dt-\sum _{i=1}^{n}f\left(w_i\right)(x_i-x_{i-1})|<ϵ.$$
   HINT: Let $P$ be a partition for which both the step functions $k$ and $l$ of part (c) are constant on the open subintervals of $P.$ Verify that for any finer partition $Q,$ $l\left(w_i\right)\ge f\left(w_i\right)\ge k\left(w_i\right),$ and hence
   $$\sum _{i}l\left(w_i\right)(x_i-x_{i-1})\ge \sum _{i}f\left(w_i\right)(x_i-x_{i-1})\ge \sum _{i}k\left(w_i\right)(x_i-x_{i-1}).$$