Real Analysis - August 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

Prove or give a counterexample: if $f_t:\mathbb R\to\mathbb R$ is a Borel measurable function for every $t\in\mathbb R$ and $f:\mathbb R\to(-\infty,+\infty]$ is defined by

$$f(x)=\sup\{f_t(x):t\in\mathbb R\},$$

then $f$ is Borel measurable.

Problem 2

Let $(X,S,\mu)$ be a measure space with finite measure, $f:X\to\mathbb R$ be a measurable function, and $f_n:X\to\mathbb R$ be a sequence of measurable functions. Suppose that $f_n\to f$ in measure. Show that $f_n^2\to f^2$ in measure.

Problem 3

Suppose that $f\in L^p([0,1])$ for some $p>1$ (with respect to the standard Lebesgue measure). Show that

$$\lim_{t\to0^+}t^{-1+\frac1p}\int_0^t f(x)\,dx=0.$$

Problem 4

Let $(X,S,\mu)$ be a measure space. Let $E_n\in S$ be a sequence of measurable subsets, and

$$A=\{x\in X:x\in E_n\text{ for infinitely many values of }n\}.$$

Suppose that $\mu(E_n)\to0$ as $n\to+\infty$. Does it always imply that $\mu(A)=0$? Prove it or give a counterexample.

Problem 5

Let $(X,S,\mu)$ be a measure space with finite measure, and $f:X\to[0,+\infty]$ be a measurable function, satisfying

$$\mu(\{x\in X:f(x)>\lambda\})<\frac1{(1+\lambda)^2}$$

for all $\lambda>0$. Show that $f\in L^1(X,\mu)$.