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\title{MTH436 - Homework 7}
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\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 4/11/18.
 
 
  
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 \section{Warm-up Questions}
 
 \begin{que}
  Let $f \in \Rint[a,b]$ and $c \in [a,b]$. Define $F_c(z) = \int_c^z f$ for all $z \in [a,b]$. Find a formula for $F_c$ in terms of $F_a$.
 \end{que}

 \begin{que}
  Calculus time! Let $f \in \Rint[a,b]$ and define $F(x) = \int_a^x f$ for all $x \in [a,b]$. Find the following in terms of $F$.
  \begin{enumerate}
   \item $G(x) = \int^b_x f$.
   \item $S(x) = \int_a^{x^2} f$.
   \item $C(x) = \int_a^{\cos x} f$.
  \end{enumerate}

 \end{que}

 \begin{que}
  Let $f \colon \R \to \R$ be continuous and $c > 0$. Define $I_f \colon \R \to \R$ by
  \[
   I_f(x) = \int_{x-c}^{x+c} f
  \]
Show that $I_f$ is differentiable and compute a formula for $I'_f(x)$.
 \end{que}

 
 
 \begin{que}
  We will prove the Integration by Substitution formula: If $J = [\alpha,\beta]$ and $\phi : J \to \R$ is continuously differentiable on $J$, and $f \colon I \to \R$ is continuous with $I$ an interval satisfying $\phi(J) \subseteq I$ then
  \[
   \int_\alpha^\beta f(\phi(t))\cdot \phi'(t) \, \mathrm{d}t = \int_{\phi(a)}^{\phi(b)} f(x) \, \mathrm{d}x.
  \]
  \begin{enumerate}
   \item Let $F(u) = \int_{\phi(a)}^u f(x) \, \mathrm{d}x$ and $G(t) = F(\phi(t))$. Prove that $G'(t) = f(\phi(t))\cdot\phi'(t)$ for all $t \in J$.
   \item By setting $t = \beta$, deduce the result.
  \end{enumerate}

 \end{que}

 
 
 \begin{que} 
  Let $\mu^*$ be outer measure on $\R$.
  \begin{enumerate}
  \item If $c \in \R$ and $A+c = \{y \in \R \mid y - c \in A\}$ satisfies $\mu^*(A) = \mu^*(A+c)$.
   \item Show that if $A \subseteq B$ then $\mu^*(A) \leq \mu^*(B)$.
   \item Show that $\mu^*(A \cup B) \leq \mu^*(A) + \mu^*(B)$. In particular, show that the union of two null sets is also null.
   \item Show that if $Z_n$ $(n\in \N)$ are null sets then $\bigcup_{n \in \N} Z_n$ is null.
  \end{enumerate}

  
 \end{que}

 \begin{que}
  Let $f,g \in \Rint[a,b]$.
  \begin{enumerate}
   \item Show that if $t \in \R$ then $\int_a^b (tf + g)^2 \geq 0$
   \item Show that if $t>0$ then $2\left| \int_a^b fg \right| \leq t \int_a^b f^2 + \frac{1}{t} \int_a^b g^2$.
  \end{enumerate}

 \end{que}

 
 
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 \section{Submitted Questions}
 
   
  \begin{que}
  Let $f \colon [a,b] \to \R$ be continuous and suppose that $\int_a^x f = \int_x^b f$ for all $x \in [a,b]$. Prove that $f(x) = 0$ for all $x \in [a,b]$.  
 \end{que}
 
 
 \begin{que}
  Let $f,g \in \Rint[a,b]$.
  \begin{enumerate}
   \item Prove that if $\int_a^b f^2 = 0$ then $\int_a^b fg = 0$.
   \item Prove that
   \[
    \left| \int_a^b fg \right|^2 \leq \left( \int_a^b |fg| \right)^2 \leq \left( \int_a^b f^2 \right)\left( \int_a^b g^2 \right).
   \]
  \end{enumerate}
\end{que}
  


 
 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Let $f,g,h \colon [a,b] \to \R$ be continuous. Define $(f,g) = \int_0^1 fg$.
  \begin{enumerate}
   \item Prove that $(f,g) = (g,f)$.
   \item Prove that $(\alpha f,g) = \alpha(f,g)$ for all $\alpha \in \R$.
   \item Prove that $(f+h,g) = (f,g) + (h,g)$.
   \item Prove that $(f,f) = 0$ if and only if $f(x) = 0$ for all $x \in [0,1]$.
  \end{enumerate}
Then prove that $\|f\| = \sqrt{(f,f)}$ defines a norm on the space of continuous functions on $[0,1]$. Also show that we have
\[
 \|f\|^2 + \|g\|^2 = \frac{\|x+y\|^2 + \|x-y\|^2}{2}
\]

 \end{que}

 \begin{que}
  We will show that outer measure is not additive. That is, it is not necessarily true that if $A$ and $B$ are disjoint then $\mu^*(A \cup B) = \mu^*(A) + \mu^*(B)$. We assume to the contrary.
  \begin{enumerate}
   \item For $x,y \in [0,1]$, let $x \sim y$ if and only if $x - y \in \Q$. Prove that $\sim$ is an equivalence relation on $[0,1]$.
   \item For each equivalence class $X$ of $\sim$, choose a representative $x \in X$. Let $A \subseteq [0,1]$ be the set of all these representatives. Show that if $q,r \in \Q$ then $(A+q) \cap (A+r) = \varnothing$.
   \item Let $Y = \Q \cap [-1,1]$. Show that
   \[
    [0,1] \subseteq U := \bigcap_{y \in Y} (A + y) \subseteq [-1,2].
   \]
   \item Show that if $\mu^*(A) = 0$ then $\mu^*(U) = 0$.
   \item Show that if $\mu^*(A) > 0$ then $\mu^*(U) = \infty$.
   \item Deduce from the previous two parts that outer measure cannot be additive.
  \end{enumerate}

 \end{que}

 
 
 \end{document}