Integration: Exercises BR Ch 11

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 2 April 2011

Integration: Exercises BR Ch 11

1. If $f\ge 0$ and ${\int }_{E}fd\mu =0$, prove that $f\left(x\right)=0$ almost everywhere on $E$. Hint: Let $E_n$ be the subset of $E$ on which $f\left(x\right)>1/n$. Write $A=\bigcup E_n$. Then $\mu \left(A\right)=0$ if and only if $\mu \left(E_n\right)=0$ for every $n$.
2. If ${\int }_{A}fd\mu =0$ for every measurable subset $A$ of a measurable set $E$, then $f\left(x\right)=0$ almost everywhere on $E$.
3. If $\left\{f_n\right\}$ is a sequence of measurable functions, prove that the set of points $x$ at which $\left\{f_n\right(x\left)\right\}$ converges is measurable.
4. If $f\in L^1\left(\mu \right)$ on $E$ and $g$ is bounded and measurable on $E$, then $fg\in L^1\left(\mu \right)$ on $E$.
5. Put

   $g\left(x\right)=\{\begin{array}{cc}0,& \text{if}0\le x\le \frac{1}{2},\\ 1,& \text{if}\frac{1}{2}\le x\le 1,\end{array}$

   and

   $f_{2k}\left(x\right)=g\left(x\right),\text{for}0\le x\le 1,\text{and}{f}_{2k+1}\left(x\right)=g(1-x),\text{for}0\le x\le 1.$

   Show that

   ${\displaystyle \underset{n\to \infty }{\mathrm{liminf}}}{f}_n\left(x\right),\text{for}0\le x\le 1,$

   but

   ${\int }_0^1{f}_n\left(x\right)dx=\frac{1}{2}$.

6. Let

   $f_n\left(x\right)=\{\begin{array}{cc}\frac{1}{n},& \text{if}\left|x\right|\le n,\\ 0,& \text{if}\left|x\right|>n.\end{array}$

   Then $f_n\left(x\right)\to 0$ uniformly on ${ℝ}^1$, but

   ${\int }_{-\infty }^{\infty }{f}_{n}dx=2,\text{for}n\in {\mathbb{Z}}_{>0}$.

   Thus uniform convergence does not imply dominated convergence in the sense of Theorem 11.32. However on sets of finite measure, uniformly convergent sequences of bounded functions do satisfy Theorem 11.32.
7. Find a necesary and sufficient condition that $f\in ℛ\left(\alpha \right)$ on $[a,b]$. Hint: Consider Example 11.6(b) and Theorem 11.33.
8. If $f\in ℛ$ on on $[a,b]$ and if $F\left(x\right)={\int }_a^{x}f\left(t\right)dt$, prove that $F\prime \left(x\right)=f\left(x\right)$ almost everywhere on $[a,b]$.
9. Prove that the function $F$ given by (96) is continuous on $[a,b]$.
10. If $\mu \left(X\right)<\infty$ and $f\in L^2\left(\mu \right)$, prove that $f\in L^1\left(\mu \right)$. If $\mu \left(X\right)=\infty$ this is false. For instance, if $m$ is Lebesgue measure on ${ℝ}^1$ and

    $f\left(x\right)={\displaystyle \frac{1}{1+\left|x\right|}},\text{for}x\in ℝ$,

    then $f\in L^2\left(m\right)$, but $f\notin L^1\left(m\right)$.
11. If $f,g\in L^1\left(\mu \right)$, define the distance between $f$ and $g$ by

    ${\int }_X|f-g|d\mu$.

    Prove that $L^1\left(\mu \right)$ is a complete metric space.
12. Suppose

    (a)   $\left|f\right(x,y\left)\right|\le 1$, if $0\le x\le 1$, $0\le y\le 1$,

    (b)   for fixed $x$, $f(x,y)$ is a continuous function of $y$,

    (c)   for fixed $y$, $f(x,y)$ is a continuous function of $x$.

    Put

    $g\left(x\right)={\int }_0^{1}f(x,y)dy$,    for $0\le x\le 1$.

    Is $g$ continuous?

13. Consider the functions

    $f_n\left(x\right)=\sin nx$,    for $n\in {\mathbb{Z}}_{>0}$ and $-\pi \le x\le \pi$,

    as points of $L^2\left(m\right)$. Prove that the set of these points is closed and bounded, but not compact.
14. Prove that a complex function $f$ is measurable if and only if $f^{-1}\left(V\right)$ is measurable for every open set $V$ in the plane.
15. Let $ℛ$ be the ring of all elementary subsets of $(0,1]$. If $0<a\le b\le 1$, define

    $\varphi \left(\right[a,b\left]\right)=\varphi \left(\right[a,b\left)\right)=\varphi \left(\right(a,b\left]\right)=\varphi \left(\right(a,b\left)\right)=b-a$,

    but define

    $\varphi \left(\right(0,b\left)\right)=\varphi \left(\right(0,b\left]\right)=1+b$,

    if $0<b\le 1$. Show that this gives an additive set function $\varphi$ on $ℛ$, which is not regular and which cannot be extended to a countably additive set function on a σ-ring.
16. Suppose $\left\{n_k\right\}$ is an increasing sequence of positive integers and $E$ is the set of all $x\in (-\pi ,\pi )$ at which $\left\{\sin n_{k}x\right\}$ converges. Prove that $m\left(E\right)=0$. Hint: For every $A\subseteq E$,

    ${\int }_A\sin n_{k}xdx\to 0$,

    and

    $2{\int }_A{\left(\sin n_{k}x\right)}^{2}dx={\int }_A(1-\cos 2n_{k}x)dx\to m\left(A\right)$      as $n\to \infty$.

17. Suppose $E\subseteq (-\pi ,\pi )$, $m\left(E\right)>0$, $\delta >0$. Use the Bessel inequality to prove that there are at most finitely many integers $n$ such that $\sin nx\ge \delta$ for all $x\in E$.
18. Suppose $f\in L^2\left(\mu \right)$ and $g\in L^2\left(\mu \right)$. Prove that

    ${\left|\int f\bar{g}d\mu \right|}^2=\int {\left|f\right|}^{2}d\mu \int {\left|g\right|}^{2}d\mu$

    if and only if there is a constant $c$ such that $g\left(x\right)=cf\left(x\right)$ almost everywhere. (Compare Theorem 11.35.)

Notes and References

These exercises are taken from [RuB, Chapt. 11] for a course in "Measure Theory" at the Masters level at University of Melbourne.

References

[RuB] W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976. MR??????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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