REAL ANALYSIS GENERAL EXAM AUGUST 18, 2025

• Solve the following five questions. The exam is 2 hours long.

• To receive full credit, you must clearly state which theorems you are using in your proofs, and your proofs must be fully justified and explained.

Question 1

Let $m$ be the Lebesgue measure on $\mathbb{R}$ and let $E$ be a Lebesgue measurable set such that $m(E)<\infty$. Let $f_{n}, f: E \rightarrow \mathbb{R}$ be measurable functions such that $f_{n} \rightarrow f$ almost everywhere. Assume that

$$\int_{E}\left|f_{n}\right|^{3} d m \leq 1, \text { for any } n \in \mathbb{N} .$$

Prove that

$$\lim _{n \rightarrow \infty} \int_{E} f_{n} d m=\int_{E} f d m$$

Question 2

Let $(X, \mathcal{M}, \mu)$ be a measure space and let $f: X \rightarrow[0, \infty]$ be a non-negative $\mathcal{M}$-measurable function on $X$. Let

$$T=\left\{t \in[0, \infty]: \mu\left(f^{-1}(\{t\})\right)>0\right\} .$$

Prove that if $f$ is integrable, then $T$ is a countable set.

Question 3

Let $\mathcal{M}$ denote the Lebesgue measurable sets on $\mathbb{R}$ and let $m$ be the Lebesgue measure on $\mathcal{M}$ and $\tau$ be the counting measure on $\mathcal{M}$. Show that $m \ll \tau$ but $\frac{d m}{d \tau}$ doesn't exist. Explain why this doesn't contradict with the Radon-Nikodym Theorem?

Question 4

Let ( $X, \mathcal{M}, \mu$ ) be a $\sigma$-finite measure space with $\mu(X)=\infty$.
(a) Show that there exists a disjoint sequence $\left\{E_{n}\right\}_{n \in \mathbb{N}} \subset \mathcal{M}$ (i.e., $E_{n} \cap E_{m}=\phi$, when $n \neq m)$, such that $\cup_{n \in \mathbb{N}} E_{n}=X$ and $\mu\left(E_{n}\right) \in[1, \infty)$ for every $n \in \mathbb{N}$.
(b) Show that there exists an $\mathcal{M}$-measurable function $f: X \rightarrow[0, \infty]$ such that $f \in L^{p}(X, \mathcal{M}, \mu)$ for all $p \in(1, \infty]$ and $f \notin L^{1}(X, \mathcal{M}, \mu)$.

Question 5

(a) Let $f \in L^{1}\left(\mathbb{R}^{n}, m\right)$. Prove that the set of points where the maximal function is finite, $\left\{x \in \mathbb{R}^{n}: M f(x)<\infty\right\}$, has full measure (i.e., its complement is a set of measure zero). The Hardy-Littlewood maximal function is defined as:

$$M f(x)=\sup _{r>0} \frac{1}{m(B(x, r))} \int_{B(x, r)} f(y) d m(y) .$$

(b) Give the definition of the Lebesgue set and state the Lebesgue Differentiation Theorem. Give an example of a function $f$ that is in $L_{\text {loc }}^{1}(\mathbb{R}, m)$ but whose Lebesgue set is not all of $\mathbb{R}$. Justify your answer.