REAL ANALYSIS GENERAL EXAM FALL 2022

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems.

Problem 1.

Let $(X, \mu)$ be a measure space, and for $n \in \mathbb{N}$, let $f_{n}: X \rightarrow[0,+\infty)$ be measurable and satisfy $\int f_{n} d \mu=1$.
(a) Show that $\int \sum_{n=1}^{\infty} n^{-2} f_{n} d \mu<+\infty$.
(b) Use the previous part to show that for almost every $x \in X$, we have that

$$\lim _{n \rightarrow \infty} \frac{f_{n}(x)}{n^{2}}=0$$

Problem 2.

Let $f \in L^{1}([0,1])$.
(a) Show that $\int_{0}^{1} \int_{x}^{1} \frac{|f(t)|}{t} d t d x<+\infty$.
(b) Show that $\int_{x}^{1} \frac{|f(t)|}{t} d t<+\infty$ for almost every $x \in[0,1]$ and that if $g(x)= \int_{x}^{1} \frac{f(t)}{t} d t$, then $g \in L^{1}([0,1])$ and $\int_{0}^{1} g(x) d x=\int_{0}^{1} f(x) d x$.

Problem 3.

Let $(X, \mu),(Y, \nu)$ be $\sigma$-finite measure spaces. Suppose that

$$B: L^{2}(X, \mu) \times L^{2}(Y, \nu) \rightarrow \mathbb{C}$$

satisfies the following:

• $f \mapsto B(f, h), g \mapsto B(k, g)$ are linear for all $h \in L^{2}(Y, \nu), k \in L^{2}(X, \mu)$.

• there is a $C \geq 0$ so that $|B(f, g)| \leq C\|f\|_{2}\|g\|_{2}$ for all $f \in L^{2}(X, \mu), g \in L^{2}(Y, \nu)$.
  Show that:
  (a) for every $f \in L^{2}(X, \mu)$ there is a $T(f) \in L^{2}(Y, \nu)$ with

$$B(f, g)=\int T(f) g d \nu, \text { for all } g \in L^{2}(Y, \nu)$$

Show that such a $T(f)$ is unique up to equality almost everywhere.
(b) the map $T$ constructed in part (a) is linear, and

$$\|T(f)\|_{2} \leq C\|f\|_{2} \text { for all } f \in L^{2}(X, \mu)$$

Problem 4.

Let $X$ be a set and $\mathcal{F} \subseteq \Sigma$ be $\sigma$-algebras of subsets of $X$. Let $\mu: \Sigma \rightarrow[0,1]$ be a probability measure.
(a) Given $f \in L^{1}(X, \Sigma, \mu)$ show that there is $g \in L^{1}(X, \mathcal{F}, \mu)$ (i.e. a $g: X \rightarrow \mathbb{C}$ which is $L^{1}$ and $\mathcal{F}$-measurable) so that

$$\int_{E} f d \mu=\int_{E} g d \mu$$

for all $E \in \mathcal{F}$. If $\widetilde{g} \in L^{1}(X, \mathcal{F}, \mu)$ is another such function, show that $g=\widetilde{g}$ almost everywhere.
(b) For $f \in L^{1}(X, \Sigma, \mu)$, denote the $g$ in part (a) by $\mathbb{E}_{\mathcal{F}}(f)$. Show that for all $h \in L^{\infty}(X, \mathcal{F}, \mu), f \in L^{1}(X, \Sigma, \mu)$ we have

$$\int f h d \mu=\int \mathbb{E}_{\mathcal{F}}(f) h d \mu$$

Problem 5.

Let $E \subseteq[0,1]$ be a Borel set with $0<m(E \cap[a, b])<b-a$ for all $0 \leq a< b \leq 1$ (you may assume the existence of such a set for this problem). Define $f:[0,1] \rightarrow[0,1]$ by $f(x)=m(E \cap[0, x])$ (here $m$ is Lebesgue measure). Show that $f$ is continuous, strictly increasing, and that for almost every $x \in E^{c}$ we have that $f^{\prime}(x)$ exists and is 0 . (It will be helpful to use the Lebesgue differentiation theorem).