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\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}