Real Analysis - August 2025

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

Let $f:\mathbb R\to\mathbb R$ be a function, not necessarily measurable. Show that

$${x\in\mathbb R:f\text{ is continuous at }x}$$

is a Borel subset of $\mathbb R$. Hint: one can actually show that it is a countable intersection of open sets.

Problem 2

For each pair of spaces:

1. $L^2([0,1])$ and $L^3([0,1])$ with the standard Lebesgue measure;

2. $L^2(\mathbb R)$ and $L^3(\mathbb R)$ with the standard Lebesgue measure;

3. $\ell^2(\mathbb Z)$ and $\ell^3(\mathbb Z)$ with the standard counting measure,

is one of them is included in the other? For example, $\ell^2(\mathbb Z)\subset\ell^3(\mathbb Z)$ or $\ell^3(\mathbb Z)\subset\ell^2(\mathbb Z)$? Provide the proof of each corresponding inclusion, or a proof that neither is included in the other. Use the usual conventions about identifying functions coinciding on zero measure sets.

Problem 3

Let $f:\mathbb R\to\mathbb R$ be a Lipschitz continuous function. Show that the graph of $f$ has measure zero in $\mathbb R^2$.

Problem 4

Let $(X,B,\mu)$ be a measure space with $\mu(X)<+\infty$, and $f_n:X\to[0,+\infty)$ be measurable functions. Show that $f_n\to0$ in measure if and only if

$$\int_X\frac{f_n^2}{1+f_n^2}\,d\mu\to0,\text{ as }n\to\infty.$$

Problem 5

Let $(X,S,\mu)$ be a measure space with $\mu(X)<+\infty$, and $f:X\to(0,+\infty)$ be a strictly positive measurable function. Show that for every $\varepsilon>0$ there exists $\delta=\delta(\varepsilon,f)>0$ such that, for every measurable subset $E\subset X$ with $\mu(E)\ge\varepsilon$, we have

$$\int_E f(x)\,dx\ge\delta.$$

Hint: consider the sets $\{x\in X:f(x)\ge1/n\}$.