I’m gonna be sick

Definition 1. A characteristic function or indicator function of a set $E$ is defined as $\chi_E(x):= \begin{cases} 1 & x\in E \\ 0 & x\notin E \end{cases}$

Definition 2. A simple function is a function which takes finitely many values. Thus for any simple function there is a set of values $a_1, a_2\dots , a_n$ and a collection of sets $E_1,E_2,\dots E_n$ such that $s(x)= \sum^n_{i=1}a_i\chi_{E_i}(x)$

Definition 3 (Lebesgue Integration). Suppose $s$ a simple function such that $s(x)= \sum^n_{i=1}a_i\chi_{E_i}(x)$ and $\mu$ is a measure on $X$ . We define $\int s \ d\mu = \sum^n_{i=1} a_i\mu(E_i).$

For a non-negative measureable function $f:X\to \mathbb R$ , we define $\int f \ d\mu = \sup\left\{\int s \ d\mu : s\le f, s \text{ is simple}\right\}$

Proposition 1 (integration over a set and useful consequences). We define $\int_Af \ d\mu :=\int f\chi_A \ d\mu$ .
Note that any characteristic function is a simple function and so for any $A$ with $\mu (A)< \infty$ we find $\mu(A) = \int \chi_A \ d \mu = \int_A 1 \ d\mu.$

Theorem 2. If $f$ is Riemann integrable, then $f$ is Lebesgue integrable and the integrals agree.

Standard Limit Theorems

Theorem 3 (Lebesgue’s Monotone Convergence Theorem). Suppose $(X,\mathcal A, \mu)$ is a measure space and $\{f_n\}_{n=0}^\infty$ is a family of real-valued measurable functions with $0\le f_1(x)\le f_{2}(x)\le \cdots$ for all $x$ and $\lim_{n\to \infty}f_n(x)=f$ for all $x$ . Then $\lim_{n\to \infty}\int f_n \ d\mu =\int f \ d \mu.$

Theorem 4 (Fatou’s Lemma). Suppose $(X,\mathcal A, \mu)$ is a measure space and $\{f_n\}_{n=0}^\infty$ is a family of real-valued nonnegative measurable functions. Then $\int\liminf_{n\to \infty} f_n \ d\mu \le \int\liminf_{n\to \infty} f_n \ d \mu.$

Theorem 5. (Dominated Convergence Theorem)Suppose $(X,\mathcal A, \mu)$ is a measure space and $\{f_n\}$ is a family of measurable real-valued functions where $f_n(x) \to f(x)$ pointwise. If there exists a non-negative integrable function $g$ such that $|f_n(x)|\le g(x)$ for all $x$ for each $n$ . Then $\int f_n \ d\mu\to \int f\ d\mu .$

Differentiation

Definition 4 (Absolutely Continuous). Let $(X,\mathcal A)$ be a measurable space and let $\mu, \nu$ be measures on $\mathcal A$ . If $\mu(E)=0\implies\nu(E)=0$ for all $E\in \mathcal A$ , then we say $\nu$ is absolutely continuous with respect to $\nu$ , denoted $\nu \ll \mu$ .

Theorem 6 (Radon-Nikodym). Suppose $\mu, \nu$ are measures on a $\sigma$ -algebra $\mathcal A$ such that $\mu$ is a $\sigma$ -finite positive measure and $\nu$ a finite positive measure with $\nu \ll \mu$ . Then there exists a $\mu$ -integrable function $f$ measurable with respect to $\mathcal A$ such that $\nu(A)=\int_Af \ d \mu$ unique up to almost everywhere equivalence.

Other

Theorem 7 (Hölder’s inequality). For functions $f,g$ and $1< p,q< \infty$ such that $\frac{1}{p}+\frac{1}{q}=1$ then $\int|fg|\ d\mu \le \|f\|_p\|g\|_q$

Lemma 9 (Chebyshev’s Inequality). If $1\le p< \infty$ , then $\mu(\{x: |f(x)|>1\})\le \frac{\int |f|^p \ d \mu }{a^p}$

Theorem 10 (Egorov’s Theorem). Suppose $\mu$ is a finite measure, $\varepsilon> 0$ , and $f_n\to f$ almost everywhere. There exists a measureable set $A$ such that $\mu(A)<\varepsilon$ and $f_n\to f$ uniformly on $A^c$

Complex

Theorem 11 (Liouville’s Theorem). If $f:\mathbb C\to \mathbb C$ is an entire function and there exists a $K$ such that $f(z)\le K$ for all $z\in \mathbb C$ then $f$ is constant

Theorem 12 (Morera’s Theorem). A function $f:U\to \mathbb C$ is holomorphic on $U$ if and only if $\int_Tf(z)\ dz=0$ for any triangle $T$ contained by $U$ .

Theorem 13. For a function $f:U\to \mathbb C$ and a closed, piecewise $C^1$ curve $\gamma$ $\frac{1}{2\pi i}\int_\gamma f(z) \ dz =$

Theorem 14 (Riemann Mapping Theorem). Every simply connected open subset of $\mathbb C$ is conformally equivalent to the open unit disk.

Theorem 15 (Riemann’s Removable Singularity Theorem). Suppose a function $f:U\to \mathbb C$ has an isolated singularity at $z_0$ . There exists a $\varepsilon> 0$ and a $K$ such that $|f(z)|<K$ for all $z\in D(z_0,\varepsilon)$ if and only if $z_0$ is a removable singularity

Proposition 16 (Cauchy’s Estimates). Suppose $f\in H(U)$ . If there exists a $\varepsilon>0$ such that $|f(z)|<M$ in a disk $D(z_0, \varepsilon)$ , then $|f^{(n)}(z_0)|\le \frac{Mn!}{\varepsilon^n}$ for $n=0,1,2, \dots$

Problems and Solutions

Real analysis

Question 1. Define $f:\mathbb R^2\to \mathbb R^2$ by $f(x,y):=(x^4-y^3, x^2+y)$ and $H:=\{(x,y)\in \mathbb R^2 : x>0 \}$ . Show that for every $(a,b)\in H$ , there is an open neighborhood $U$ of $(a,b)$ with $U\subset H$ , on which $f$ is injective, and there is a differentiable $g:f(U)\to U$ such that $g(f(x,y))=(x,y)$ for all $(x,y)\in U$

Answer 1. Set $u(x,y)=x^4-y^3$ and $v(x,y)=x^2+y$ so that $f=(u,v)$ . (Scratch: Taking all of the first order partial derivatives $u_x=4x^3$ , $u_y=-3y^2$ , $v_x=2x$ and $v_y=1$ ) . For all $(x,y)\in \mathbb R^2$ , the determinant of the Jacobian of $f$ at $(x,y)$ is $4x^3+6y^2x=2x(2x^2+6y^2)$ . Given any $(a,b)\in H$ , since $a>0$ by construction and $b^2\ge 0$ for any $b$ , $2a(2a^2+6b^2)\ne 0$ . Therefore, the Inverse Function Theorem states that there is such a neighborhood $U$ of $(a,b)$ and such a function $g:f(U)\to U$ . 0◻

Question 2. For $f\in L^1(\mathbb R, \mathcal L , m)$ and $t\in \mathbb R$ define $\widehat{f}(t):=\int _\mathbb R f(x)\exp(-2\pi ixt) \ dm(x).$ Show that $\widehat{f}(x)$ is a continuous function of $t$ .

Question 3. Let $(X, \mathcal M, \mu )$ be a measure space and $f\in L^1(X, \mathcal M, \mu)$ . Show that, for all $\varepsilon > 0$ , there is a $\delta>0$ so that, for all $E \in \mathcal M$ , $\mu(E) < \delta \implies \int_E |f | d\mu < \varepsilon.$

Answer 2. Since $f\in L^1(X,\mathcal M, \mu)$ , it must be the case $|f|$ a non-negative measurable functions. Thus, there is a sequence of non-negative simple functions bounded above by $|f|$ and converging to $|f|$ pointwise. Given $\varepsilon>0$ we can choose some simple function $s$ such that $\int |f|\ d\mu \le \int s \ d\mu + \varepsilon$ .

Question 4. Let $(X, \mathcal M, \mu )$ be a measure space. A family $\mathcal F$ of measurable functions is said to be uniformly integrable if, for every $\varepsilon > 0$ , there is a $\delta>0$ so that, for all $E \in \mathcal M$ and $f\in \mathcal F,$ $\mu(E) < \delta \implies \int_E |f | d\mu < \varepsilon.$ Show that any finite $\mathcal F\subset L^1(X,\mathcal M, \mu )$ is uniformly integrable. (You may ignore $\mathcal F=\emptyset$ )

4. Let

D = \{z \in \mathbb C \mid |z| < 1\}

be the unit disc. Either construct a holomorphic function

f: D \to D

with

f(1/2) = 3/4

and

f'(1/2) = 2/3

or prove that one does not exist.
5. Set

\mathbb D = \{z \in \mathbb C : |z| < 1\}

and

\Omega = \mathbb D \setminus \big((-1, -1/2] \cup [1/2, 1)\big)

. Give an explicit conformal equivalence

f: \Omega \to \mathbb H