\documentclass[12pt,twoside,reqno]{amsart} \usepackage{amssymb,amsmath,amstext,amsthm,amsfonts,amscd, accents} \usepackage{amsthm} \theoremstyle{plain} \usepackage[ansinew]{inputenc} \usepackage{graphicx} \usepackage[mathscr]{eucal} \usepackage{hyperref} %\usepackage{bbm} \usepackage{dsfont} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\T}{\mathbb{T}} \newcommand{\E}{\mathbb{E}} \newcommand{\B}{\mathcal{B}} %\newcommand{\P}{\mathbb{P}} \newcommand{\comp}{^{\complement}} \newcommand{\card}{\rm{card \,}} \newcommand{\abs}[1]{\bigl| #1 \bigr|} % absolute value \newcommand{\sabs}[1]{\left| #1 \right|} % smaller absolute value \newcommand{\babs}[1]{\Bigl| #1 \Bigr|} % big absolute value \newcommand{\norm}[1]{\lVert#1\rVert} % norm \newcommand{\bnorm}[1]{\Bigl\| #1\Bigr\|} %big norm \newcommand{\avg}[1]{\left< #1 \right>} % average \newcommand{\mstar}{m^\ast} % outer measure \newcommand{\less}{\lesssim} \newcommand{\more}{\gtrsim} \newcommand{\ep}{\epsilon} \newcommand{\setdif}{\bigtriangleup} \newcommand{\nzerobar}{\underline{n_0}} \newcommand{\Elem}{\mathscr{E}} \newcommand{\Jordan}{\mathscr{J}} \newcommand{\dist}{{\rm dist}} %\newcommand{\ind}{\mathds{1}} \newcommand{\ind}{\textbf{1}} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition} \newtheorem{corollary}[proposition]{Corollary} \newtheorem{lemma}[proposition]{Lemma} \newtheorem*{lemma*}{Lemma} \newtheorem{definition}{Definition} \newtheorem*{definition*}{Definition} \newtheorem{problem}{Problem} \let\oldproblem\problem \renewcommand{\problem}{\oldproblem\normalfont} %\newtheorem{problem}{Problem} \newtheorem*{hypothesis}{Hypothesis} \newtheorem{exercise}{Exercise}[section] \newtheorem{remark}{Remark}[section] \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \usepackage{enumerate} \newcommand{\figura}[2]{ \begin{center} \includegraphics*[scale={#2}]{{#1}.pdf} \end{center} } \title[Review problems]{Review problems\\TMA4225-Foundations of Analysis-Fall 2015} %\author{Silvius Klein} \begin{document} \maketitle \begin{problem} Let $p (x)$ be a polynomial function. Prove that the set $$\{ x \in [0, 1] \colon \, p (x) \le 0 \}$$ is {\em Jordan} measurable. \textit{Hint:} Consider the boundary of this set. \smallskip Is the same true for continuous functions? How about Lebesgue measurability? \end{problem} %%%% \medskip \begin{problem} Let $(X, \B, \mu)$ be a measure space, let $f \colon X \to \R$ be an absolutely integrable function and let $M$ and $m$ be two real numbers. \begin{enumerate}[a)] \item Prove that if $f (x) \le M$ for a.e. $x \in X$, then $$\int_X f \, d \mu \le M \cdot \mu (X) .$$ \item Prove that if $f (x) \ge m$ for a.e. $x \in X$, then $$\int_X f \, d \mu \ge m \cdot \mu (X) .$$ \end{enumerate} \end{problem} %%%% \medskip \begin{problem} Let $f \colon [a, b] \to \R$ be a continuous function. Prove that $f$ is Lebesgue absolutely integrable and that $$\int_a^b f (x) d x = \int_{[a, b]} f (x) d x \, ,$$ where the left hand side refers to the Riemann-Darboux integral of $f$, while the right hand side to the Lebesgue integral of $f$. \smallskip To make it easier, divide the problem into the following steps: \begin{enumerate}[a)] \item Explain why $f$ lives in a box". \item Prove that $\int_a^b f (x) d x \le \int_{[a, b]} f (x) d x$ by using approximations by step / simple functions from below. \item Prove that $\int_a^b f (x) d x \ge \int_{[a, b]} f (x) d x$ by using approximations by step / simple functions from above. \end{enumerate} \end{problem} %%%% \medskip \begin{problem} Compute the following limit $$\lim_{n\to\infty} \, \int_0^1 \, n \, x^2 \, \sin \left(\frac{1}{n x} \right) \, d x \, .$$ \smallskip \textit{Hint:} It helps to remember that $$\lim_{t \to 0} \, \frac{\sin t }{t} = 1 \, .$$ \end{problem} %%%% \medskip \begin{problem} Compute the limit $$\lim_{n\to\infty} \, \int_0^1 \, \sin \left(\frac{x}{n} \right) \cdot \ln x \, d x \, .$$ To make it easier, we may divide the problem into: \begin{enumerate}[a)] \item Prove that the function $g (x) := \ln x$ is absolutely integrable on $[0, 1]$. \item For every $n \ge 1$ consider the function $f_n (x) := \sin \left(\frac{x}{n} \right) \cdot \ln x$. \\Explain why these functions are Lebesgue measurable on $[0, 1]$. \item Compute the pointwise limit of this sequence of functions and then apply the appropriate convergence theorem. \end{enumerate} \end{problem} %%%% \medskip \begin{problem} Let $f \colon \R \to \R$ be an absolutely integrable function, let $a$ be a fixed real number and define the function $$F (x) := \int_{[a, x]} \, f (t) \, d t \, .$$ Prove that $F$ is continuous everywhere. \smallskip \textit{Hint:} Use the dominated convergence theorem. \end{problem} %%%% \medskip \begin{problem} Let $(X, \B, \mu)$ be a measure space and let $f_1, \, f_2, \, \ldots \, \colon X \to [0, \infty]$ be a sequence of measurable functions. Prove that if $$\sum_{n=1}^\infty \int_X f_n \, d \mu < \infty \, ,$$ then $$\lim_{n\to\infty} f_n (x) = 0 \quad \text{for } \mu \text{ a.e.} \ x \in X .$$ \smallskip \textit{Hint:} Use Tonelli's convergence theorem. \end{problem} %%%% \medskip \begin{problem} Let $(X, \B, \mu)$ be a measure space and let $f \colon X \to \R$ be an absolutely integrable function. For every $n \ge 1$ define the set $$E_n := \{ x \in X \colon \abs{f (x)} \ge n^2 \} \, .$$ Prove that $$\sum_{n=1}^\infty \, \mu (E_n) < \infty .$$ %\smallskip \textit{Hint:} Use Markov's inequality. \end{problem} %%%% \medskip \begin{problem} Let $f \colon \R \to \R$ be an absolutely integrable function. Prove that for all $t \in \R$ we have $$\int_\R f (x+t) \, d x = \int_\R f (x) \, d x \, .$$ \smallskip \textit{Hint:} Consider first the case when $f = \ind_E$ is an indicator function (you need to apply the translation invariance of the Lebesgue measure, homework problem 5.1). Then let $f$ be a simple function. After that let $f$ be non-negative and approximate it by simple functions. Finally, split $f$ into its positive and negative parts. \end{problem} %%%% %\smallskip \begin{problem} Let $f \colon \R \to [0, \infty)$ be a measurable function and let $$A := \{ (x, y) \in \R^2 \colon x \in \R \ \text{ and } 0 < y < f (x) \}$$ be the region under the graph of $f$. Prove that $A$ is a Lebesgue measurable set in $\R^2$ and that its measure is the Lebesgue integral of $f$, that is $$m (A) = \int_\R f (x) \, d x .$$ \smallskip \textit{Hint:} First verify these statements when $f$ is an indicator function: $f = \ind_E$, with $E$ Lebesgue measurable in $\R$. Then do it for a simple function. Finally, for the general case, consider an approximation by an increasing sequence of simple functions and apply the monotone convergence theorem. \end{problem} %%%% %\medskip \begin{problem} Let $(X, \B)$ be a measurable space. Let $\mu_1$ and $\mu_2$ be two measures on $(X, \B)$ and define $\mu := \mu_1 + \mu_2$. More precisely, for every $A \in \B$, let $$\mu (A) := \mu_1 (A) + \mu_2 (A) \, .$$ \begin{enumerate}[a)] \item Prove that $\mu$ is a measure on $(X, \B)$. \item Prove that for every measurable function $f \colon X \to [0, \infty]$, $$\int_X f \, d \mu = \int_X f \, d \mu_1 + \int_X f \, d \mu_2 \, .$$ \end{enumerate} \end{problem} %\bigskip % %The next problems are a bit more tricky, take them as a challenge. % %\begin{problem} Let $(X, \B, \mu)$ be a measure space. %\begin{enumerate}[a)] %\item Let $f \colon X \to [0, \infty]$ be a measurable function. Prove that if for every measurable set $E$ with $\mu (E) > 0$ we have %$$\int_E f \, d \mu = 0 \, ,$$ %then $f=0$ a.e. % %\item Let $f \colon X \to \R$ be an absolutely integrable function. Prove that if for every measurable set $E$ with $\mu (E) > 0$ we have %$$\int_E f \, d \mu \ge 0 \, ,$$ %then $f \ge 0$ a.e. %\end{enumerate} % %\end{problem} % %\begin{problem} %Let $(X, \B, \mu)$ be a measure space and let $\phi \colon X \to [0, \infty]$ be a measurable function. Define the function %$F \colon [0,\infty) \to [0, \infty)$ by %$$F (t) := \mu \{ x \in X \colon f (x) > t \} \, .$$ % %Prove that $$\int_X \phi \, d \mu = \int_0^\infty F (t) \, d t \, .$$ %\end{problem} \end{document}