Real Analysis - January 2023

Note: all statements require proofs. You can make references to standard theorems from the course; however, you need state the relevant part of the theorem in your own words, unless it’s a well known named theorem. For example, ”we had a theorem in the class that said that any continuous function on a compact subset of $\mathbb{R}^n$ is uniformly continuous” is a good reference, and ”by the uniqueness theorem from the class, $f$ is unique” is not a good reference.

Problem 1

Find three sequences of Borel functions $f_n,g_n,h_n:\mathbb R\to\mathbb R$ such that:

1. $f_n\to0$ in $L^1(\mathbb R)$ but does not converge almost everywhere.

2. $g_n\to0$ in measure but converges neither in $L^1(\mathbb R)$ nor almost everywhere.

3. $h_n\to0$ almost everywhere on $\mathbb R$ but converges neither in $L^1(\mathbb R)$ nor in measure.

In all types of convergence, use the Lebesgue measure.

Problem 2

Let $f\in L^p([0,3]^2,\mu)$, where $\mu$ is the standard Lebesgue measure on $[0,3]^2\subset\mathbb R^2$ and $1<p<+\infty$. Show that

$$\int_{[0,3]^2}f\,d\mu\le C_p\|f\|_p,$$

where $C_p$ depends on $p$ but not on $f$. Find the smallest possible value of such $C_p$.

Problem 3

Let $f:\mathbb R\to\mathbb R$ be a continuous function satisfying $f(y)-f(x)\ge y-x$ for all $x,y\in\mathbb R$ with $x<y$. Show that $|f^{-1}(A)|=0$ for every $A\subset\mathbb R$ with $|A|=0$. Here $|\cdot|$ denotes the Lebesgue outer measure. Hint: note that $f$ is strictly monotone and one-to-one.

Problem 4

Let $(X,S,\mu)$ be a measure space with $\mu(X)<+\infty$. Let $f:X\to[0,+\infty)$ be a measurable function. Show

$$\int_X e^{f(x)}\,dx=\mu(X)+\int_0^{+\infty}\mu(\{x\in X:f(x)\ge s\})e^s\,ds.$$

Problem 5

Let $(X,S,\mu)$ be a measure space with a $\sigma$-finite measure. Suppose that $f_n:X\to\mathbb R$ are measurable functions, and $f_n\to f$ almost everywhere on $X$. Show that there exist measurable subsets $\{E_k\}_{k\in\mathbb N}$ such that $f_n$ converges to $f$ uniformly on $E_k$, and

$$\mu\left(X\setminus\bigcup_{n=1}^{\infty}E_n\right)=0.$$