Integration: Exercises Chapter 2

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

Last updates: 20 March 2011

Integration: Exercises Chapter 2

1. Let $\left\{f_n\right\}$ be a sequence of real nonnegative functions on ${ℝ}^1$ and consider the following four statements:

   (a)   If $f_1$ and $f_2$ are upper semicontinuous, then $f_1+f_2$ is upper semicontinuous,

   (b)   If $f_1$ and $f_2$ are lower semicontinuous, then $f_1+f_2$ is lower semicontinuous,

   (c)   If each $f_n$ is upper semicontinuous, then ${\sum }_1^{\infty }{f}_n$ is upper semicontinuous,

   (d)   If each $f_n$ is lower semicontinuous, then ${\sum }_1^{\infty }{f}_n$ is lower semicontinuous,

   Show that three of these are true and one is false. What hapens if the word "nonnegative" is omitted? Is the truth of the statements affected if ${ℝ}^1$ is replaced by a general topological space?
2. Let $f$ be an arbitrary function on ${ℝ}^1$, and define

   $\varphi (x,\delta )=\sup \left\{\right|f\left(s\right)-f\left(t\right)|s,t\in (x-\delta ,x+\delta \left)\right\}$,
   $\varphi \left(x\right)=\inf \left\{\varphi \right(x,\delta \left)\right|\delta >0\}$,

   Prove that $\varphi$ is upper semicontinuous, that $f$ is continuous at a point $x$ if and only if $\varphi \left(x\right)=0$, and hence that the set of points of continuity of an arbitrary complex function is a $G_{\delta }$.
   Formulate and prove an analogous statement for general topological spaces in place of ${ℝ}^1$.
3. Let $X$ be a metric space, with metric $\rho$. For any nonempty $E\subseteq X$, define

   ${\rho }_E\left(x\right)=\inf \left\{\rho (\right(x,y\left)\right|y\in E\}$.

   Show that ${\rho }_E$ is a uniformly continuous function on $X$. If $A$ and $B$ are disjoint nonempty closed subsets of $X$, examine the relevance of the function

   $f\left(x\right)={\displaystyle \frac{{\rho }_A\left(x\right)}{{\rho }_A\left(x\right)+{\rho }_B\left(x\right)}}$

   to Urysohn's lemma.
4. Examine the proof of the Riesz theorem and prove the following two statements:

   (a)   If $E_1\subseteq V_1$ and $E_2\subseteq V_2$, where $V_1$ and $V_2$ are disjoint open sets, then $\mu (E_1\cup E_2)=\mu (E_1+E_2)$, even if $E_1$ and $E_2$ are not in $ℳ$.

   (b)   If $E\in {ℳ}_F$, then $E=N\cup K_1\cup K_2\cup \cdots$, where $\left\{K_i\right\}$ is a disjoint countable collection of compact sets and $\mu \left(N\right)=0$.

5. Let $m$ denote Lebesgue measure on ${ℝ}^1$. Let $E$ be Cantor's familiar "middle thirds" set. Show that $m\left(E\right)=0$ even though $E$ and ${ℝ}^1$ have the same cardinality.
6. Let $m$ denote Lebesgue measure on ${ℝ}^1$. Construct a totally disconnected compact set $K\subseteq {ℝ}^1$ such that $m\left(K\right)>0$. ($K$ is to have no connected subset consisting of more than one point.)
   If $v$ is lower semicontinuous and $v\le {\chi }_K$, show that actually $v\le 0$. Hence ${\chi }_K$ cannot be approximated from below by lower semicontinuous functions, in the sense of the Vitali-Carathéodory theorem.
7. Let $m$ denote Lebesgue measure on ${ℝ}^1$. If $0<ϵ<1$, construct an open set $E\subseteq [0,1]$ which is dense in $[0,1]$, such that $m\left(E\right)=ϵ$. (To say that $A$ is dense in $B$ means that the closure of $A$ contains $B$.)
8. Let $m$ denote Lebesgue measure on ${ℝ}^1$. Construct a Borel set $E\subseteq {ℝ}^1$ such that

   $0<m(E\cap I)<m\left(I\right)$

   for every nonempty segment $I$. Is it possible to have $m\left(E\right)<\infty$ for such a set?
9. Construct a sequence of continuous functions $f_n$ on $[0,1]$ such that $0\le f_n\le 1$, such that

   ${\displaystyle \underset{n\to \infty }{\lim }{\int }_0^1{f}_n\left(x\right)dx=0}$,

   but such that the sequence $\left\{f_n\right(x\left)\right\}$ converges for no $x\in [0,1]$.
10. If $\left\{f_n\right\}$ is a sequence of continuous functions $f_n$ on $[0,1]$ such that $0\le f_n\le 1$ and such that $f_n\left(x\right)\to 0$ as $n\to \infty$ for every $x\in [0,1]$, then

    ${\displaystyle \underset{n\to \infty }{\lim }{\int }_0^1{f}_n\left(x\right)dx=0}$.

    Try to prove this without using any measure theory or any theorems about Lebesgue integration. (This is to impress you with the power of the Lebesgue integral. A nice proof was given by W.F. Oberlein in Communications on Pure and Applied Mathematics, vol. X, pp. 357-360, 1957.)
11. Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$; assume that $\mu \left(X\right)=1$. Prove that there is a compact set $K\subseteq X$ (the carrier or support of $\mu$) such that $\mu \left(K\right)=1$ but $\mu \left(H\right)<1$ for every proper comapct subset $H$ of $K$. Hint: Let $K$ be the intersection of all compact $K_{\alpha }$ with $\mu \left(K_{\alpha }\right)=1$; show that every open set $V$ which contains $K$ also contains some $K_{\alpha }$. Regularity of $\mu$ is needed; compare exercise 18. Show that $K^c$ is the largest open set in $X$ whose measure is $0$.
12. Show that every compact subset of ${ℝ}^1$ is the support of a Borel measure.
13. Is it true that every compact subset of ${ℝ}^1$ is the support of a continuous function? If not, can you describe the class of all compact sets in ${ℝ}^1$ which are supports of continuous functions? Is your description valid in other topological spaces?
14. Let $f$ be a real-valued Lebesgue measurable function on ${ℝ}^k$. Prove that there exist Borel functions $g$ and $h$ such that $g\left(x\right)=h\left(x\right)$ a.e. $\left[m\right]$, and $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$ for every $x\in {ℝ}^k$.
15. It is easy to guess the limits of

    ${\displaystyle {\int }_0^n{(1-\frac{x}{n})}^n{e}^{x/2}dx}$       and       ${\displaystyle {\int }_0^n{(1+\frac{x}{n})}^n{e}^{-2x}dx}$

    as $n\to \infty$. Prove that your guesses are correct.
16. Why is $m\left(Y\right)=0$ in the proof of Theorem 2.20(e)?
17. Define the distance between points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane to be

    $|y_1-y_2|$    if $x_1=x_2$,        $1+|y_1-y_2|$    if $x_1\ne x_2$.

    Show that this is indeed a metric, and that the resulting metric space $X$ is locally compact.
    If $f\in C_c\left(X\right)$, let $x_1,\dots x_n$ be those values of $x$ for which $f(x,y)\ne 0$ for at least one $y$ (there are only finitely many such $x$!), and define

    $\Lambda f={\displaystyle \sum _{j=1}^n{\int }_{-\mathrm{\infty }}^{\infty }f(x_j,y)dy}$.

    Let $\mu$ be the measure associated with this $\Lambda$ by Theorem 2.14. If $E$ is the $x$-axis, show that $\mu \left(E\right)=\infty$ although $\mu \left(K\right)=0$ for every compact $K\subseteq E$.
18. Let $X$ be a well-ordered uncountable set which has a last element ${\omega }_1$, such that every predecessor of ${\omega }_1$ has at most countably many predecessors. ("Construction": Take any well-ordered set which has elements with uncountably many predecessors, and let ${\omega }_1$ be the first of these; ${\omega }_1$ is called the first countable ordinal.) For $\alpha \in X$, let $P_{\alpha }$ (resp. $S_{\alpha }$) be the set of all predecessors (resp. successors) of $\alpha$, and call a subset of $X$ open if it is a $P_{\alpha }$ or an $S_{\beta }$ or a $P_{\alpha }\cap S_{\beta }$ or a union of such sets. Prove that $X$ is a compact Hausdorff space. (Hint: No well ordered set contains an infinite decreasing sequence.)

    (a)   Prove that the complement of the point ${\omega }_1$ is an open set which is not σ-compact.

    (b)   Prove that to every $f\in C\left(X\right)$ there corresponds an $\alpha \ne {\omega }_1$ such that $f$ is constant on $S_{\alpha }$.

    (c)   Prove that the intersection of every countable collection $\left\{K_n\right\}$ of uncountable compact subsets of $X$ is uncountable. (Hint: Consdier limits of increasing countable sequences in $X$ which intersect each $K_n$ in infinitely many points.)

    Let $ℳ$ be the collection of all $E\subseteq X$ such that either $E\cup \left\{{\omega }_1\right\}$ or $E^c\cup \left\{{\omega }_1\right\}$ contains an uncountable compact set; in the first case, define $\lambda \left(E\right)=1$; in the second case, define $\lambda \left(E\right)=0$. Prove that $ℳ$ is a σ-algebra which contains all Borel sets in $X$, that $\lambda$ is a measure on $ℳ$ which is not regular (every neighborhood of ${\omega }_1$ has measure 1), and that

    $f\left({\omega }_1\right)={\displaystyle {\int }_{X}fd\lambda }$.

    for every $f\in C\left(X\right)$. Describe the regular $\mu$ which Theorem 2.14 associates with this linear functional.
19. Go through the proof of Theorem 2.14, assuming $X$ to be compact (or even compact metric) rather than just locally compact, and see what simplifications you can find.
20. Find continuous functions $f_n:[0,1]\to [0,\infty )$ such that $f_n\left(x\right)\to 0$ for all $x\in [0,1]$ as $n\to \infty$, ${\int }_0^1{f}_n\left(x\right)dx\to 0$, but ${\sup }_n{f}_n$ is not in $L^1$. (This shows that the conclusion of the dominated convergence theorem may hold even when part of its hypothesis is violated.)
21. If $X$ is compact and $f:\to (-\infty ,\infty )$ is upper semicontinuous, prove that $f$ attains its maximum at some point of $X$.
22. Suppose that $X$ is a metric space, with metric $d$, and that $f:X\to [0,\infty ]$ is lower semicontinuous $f\left(p\right)<\infty$ for at least one $p\in X$. For $n=1,2,3,\dots$ and $x\in X$, define

    $g_n\left(x\right)=\inf \left\{f\right(p)+nd(x,p)|p\in X\}$

    and prove that

    (i)   $|g_n\left(x\right)-g_n\left(y\right)|\le nd(x,y)$,

    (ii)   $0\le g_1\le g_2\le \cdots \le f$,

    (iii)   $g_n\left(x\right)\to f_n\left(x\right)$ as $n\to \infty$, for all $x\in X$.

    Thus $f$ is the pointwise limit of an increasing sequence of continuous functions. (Note that the converse is almost trivial.)
23. Suppose that $V$ is open in ${ℝ}^k$ and $\mu$ is a finite positive Borel measure on ${ℝ}^k$. Is the function that sends $x$ to $\mu (V+x)$ necessarily continuous? lower semicontinuous? upper semicontinuous?
24. A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in ${ℝ}^1$. Assume $f\in L^1\left({ℝ}^1\right)$, and prove that there is a sequence $\left\{g_n\right\}$ of step functions so that

    ${\displaystyle \underset{n\to \infty }{\lim }{\int }_{-\infty }^{\infty }|f\left(x\right)-g_n\left(x\right)|dx=0}$.

25. (i)   Find the smallest constant $c$ such that

    $\log (1+e^t)<c+t(0<t<\infty )$.

    (ii)   Does

    ${\displaystyle \underset{n\to \infty }{\lim }\frac{1}{n}{\int }_0^1\log (1+e^{nf\left(x\right)})dx}$.

    exist for every real $f\in L^1$? If it exists, what is it?

Notes and References

These exercises are taken from [Ru, Chapt. 2] for a course in "Measure Theory" at the Masters level at University of Melbourne.

References

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

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