\documentclass[11pt]{amsart}
\usepackage{amscd,amssymb}
\theoremstyle{plain}
\newtheorem{Thm}[equation]{Theorem}
\newtheorem{Cor}[equation]{Corollary}
\newtheorem{Prop}[equation]{Proposition}
\newtheorem{Lem}[equation]{Lemma}
\newtheorem{Def}[equation]{Definition}
\numberwithin{equation}{section}

\setlength{\oddsidemargin}{0.2in}
\setlength{\evensidemargin}{0.2in}
\setlength{\textwidth}{6.1in}

\begin{document}



\title{HW II for Math 6210}
\maketitle
\noindent 
In the following exercises $X$ is always a measure space, i.e. there is a sigma algebra $\mathcal M \subset P(X)$ and a measure 
$\mu$ on it. 

\vskip 10pt 
\noindent  
1) Assume that $X=A\cup B$ where $A,B\in \mathcal M$. Let $f: X \rightarrow \mathbb R$. Prove that $X$ is measurable if and only if the restrictions of $f$ to $A$ and $B$ 
are measurable. 


\vskip 10pt 
\noindent   
2) Let $f \in L^+(X)$ such that $\int f  <\infty$. Prove that, for every $\epsilon >0$, there exists a set of finite measure $E \subset X$ such that 
\[ 
\int_E f > \int f -\epsilon. 
\] 

\vskip 10pt 
\noindent  
3) Suppose a sequence $\{f_n\}$ in $L^{+}(X)$ decreases pointwise to $f$, and $\int f_1 < \infty$. Prove that $\lim_{n\rightarrow \infty} \int f_n =\int f$. 
(Do not use the dominated convergence theorem or Fatou's lemma.) 

\vskip 10pt 
\noindent  
4) Let $f\in L^+(X)$. Then $\lambda(E)=\int_E f d\mu$ is a measure on $\mathcal M$ and for every $g\in L^+(X)$,  
$\int g d\lambda  = \int gf d\mu$. Hint: do this for simple $g$ firstly. 

\vskip 10pt 
\noindent  
5) Let  $\{f_n\} \in L^+(X)$ converging pointwise to $f$. Assume that $\int f_n \rightarrow \int f < \infty$.  Then 
$\int_E f_n \rightarrow \int _E f $  for every $E \in \mathcal M$. 

\vskip 10pt 
\noindent  
6) Prove that $f(x)=x\exp(-\frac{x^2}{2})$ is Lebesgue integrable on $[0,\infty)$ and compute its integral. 

\vskip 10pt 
\noindent 
7) Please justify all steps in the following: 
\begin{enumerate} 
\item Compute the derivative of $g(t)=\int_{\mathbb R} \exp(-\frac{x^2}{2}) \cos(tx) dx$, where $t\in \mathbb R$. 
\item Use $\exp(-\frac{x^2}{2})'=-x \exp(-\frac{x^2}{2})$ and integration by parts to show that $g(t)$ satisfies a first order differential equation. 
Solve it to find $g$. 
\end{enumerate} 





 

\end{document} 


