I’d like to extend my gratitude to Peter Woolfitt for supplying many solutions and checking many proofs of the rest in problem sessions. Many other solutions contain input and ideas from other graduate students and faculty members at UGA, along with questions and answers posted on Math Stack Exchange or Math Overflow.
Let \(f(x) = \frac 1 x\). Show that \(f\) is uniformly continuous on \((1, \infty)\) but not on \((0,\infty)\).
Let \begin{align*} f(x) = \sum _{n=0}^{\infty} \frac{x^{n}}{n !}. \end{align*}
Describe the intervals on which \(f\) does and does not converge uniformly.
Let \(\left\{{f_n}\right\}\) be a sequence of continuous functions such that \(\sum f_n\) converges uniformly.
Prove that \(\sum f_n\) is also continuous.
Let \(f(x, y)\) on \([-1, 1]^2\) be defined by \begin{align*} f(x, y) = \begin{cases} \frac{x y}{\left(x^{2}+y^{2}\right)^{2}} & (x, y) \neq (0, 0) \\ 0 & (x, y) = (0, 0) \end{cases} \end{align*} Determine if \(f\) is integrable.
Let \((X, d)\) and \((Y, \rho)\) be metric spaces, \(f: X\to Y\), and \(x_0 \in X\).
Prove that the following statements are equivalent:
Let \(I\) be an index set and \(\alpha: I \to (0, \infty)\).
Show that \begin{align*} \sum_{i \in I} a(i):=\sup _{\substack{ J \subset I \\ J \text { finite }}} \sum_{i \in J} a(i)<\infty \implies I \text{ is countable.} \end{align*}
Suppose \(I = {\mathbb{Q}}\) and \(\sum_{q \in \mathbb{Q}} a(q)<\infty\). Define \begin{align*} f(x):=\sum_{\substack{q \in \mathbb{Q}\\ q \leq x}} a(q). \end{align*} Show that \(f\) is continuous at \(x \iff x\not\in {\mathbb{Q}}\).
Let \(\left\{{a_n}\right\}\) be a sequence of real numbers such that \begin{align*} \left\{{b_n}\right\} \in \ell^2({\mathbb{N}}) \implies \sum a_n b_n < \infty. \end{align*} Show that \(\sum a_n^2 < \infty\).
Note: Assume \(a_n, b_n\) are all non-negative.
Prove that if \(f: [0, 1] \to {\mathbb{R}}\) is continuous then \begin{align*} \lim_{k\to\infty} \int_0^1 kx^{k-1} f(x) \,dx = f(1) .\end{align*}
Let \(\{a_n\}_{n=1}^\infty\) be a sequence of real numbers.
Prove that if \(\displaystyle\lim_{n\to \infty } a_n = 0\), then \begin{align*} \lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0 \end{align*}
Prove that if \(\displaystyle\sum_{n=1}^{\infty} \frac{a_{n}}{n}\) converges, then \begin{align*} \lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0 \end{align*}
Let \(f\in L^1([0, 1])\). Prove that \begin{align*} \lim_{n \to \infty} \int_{0}^{1} f(x) {\left\lvert {\sin n x} \right\rvert} ~d x= \frac{2}{\pi} \int_{0}^{1} f(x) ~d x \end{align*}
Hint: Begin with the case that \(f\) is the characteristic function of an interval.
Let \begin{align*} f_{n}(x) = n x(1-x)^{n}, \quad n \in {\mathbb{N}}. \end{align*}
Hint: Consider the maximum of \(f_n\).
Let \begin{align*} f_{n}(x) = a e^{-n a x} - b e^{-n b x} \quad \text{ where } 0 < a < b. \end{align*}
Show that
\(\sum_{n=1}^{\infty} \left|f_{n}\right|\) is not in \(L^{1}([0, \infty), m)\)
Hint: \(f_n(x)\) has a root \(x_n\).
\begin{align*} \sum_{n=1}^{\infty} f_{n} \text { is in } L^{1}([0, \infty), m) {\quad \operatorname{and} \quad} \int _{0}^{\infty} \sum _{n=1}^{\infty} f_{n}(x) \,dm = \ln \frac{b}{a} \end{align*}
Define \begin{align*} f(x) = \sum_{n=1}^{\infty} \frac{1}{n^{x}}. \end{align*}
Show that \(f\) converges to a differentiable function on \((1, \infty)\) and that \begin{align*} f'(x) =\sum_{n=1}^{\infty}\left(\frac{1}{n^{x}}\right)^{\prime}. \end{align*}
Hint: \begin{align*} \left(\frac{1}{n^{x}}\right)' = -\frac{1}{n^{x}} \ln n \end{align*}
Let \(\phi\in L^\infty({\mathbb{R}})\). Show that the following limit exists and satisfies the equality \begin{align*} \lim _{n \to \infty} \left(\int _{\mathbb{R}} \frac{|\phi(x)|^{n}}{1+x^{2}} \, dx \right) ^ {\frac{1}{n}} = {\left\lVert {\phi} \right\rVert}_\infty. \end{align*}
Let \(f, g \in L^2({\mathbb{R}})\). Show that \begin{align*} \lim _{n \to \infty} \int _{{\mathbb{R}}} f(x) g(x+n) \,dx = 0 \end{align*}
For \(n\in {\mathbb{N}}\), define \begin{align*} e_{n} = \left (1+ {1\over n} \right)^{n} {\quad \operatorname{and} \quad} E_{n} = \left( 1+ {1\over n} \right)^{n+1} \end{align*}
Show that \(e_n < E_n\), and prove Bernoulli’s inequality:
\begin{align*}
(1+x)^{n} \geq 1+n x \text { for }-1
Use this to show the following:
Define \begin{align*} f(x)=c_{0}+c_{1} x^{1}+c_{2} x^{2}+\ldots+c_{n} x^{n} \text { with } n \text { even and } c_{n}>0. \end{align*}
Show that there is a number \(x_m\) such that \(f(x_m) \leq f(x)\) for all \(x\in {\mathbb{R}}\).
Let \(m_*\) denote the Lebesgue outer measure on \({\mathbb{R}}\).
Prove that for every \(E\subseteq {\mathbb{R}}\) there exists a Borel set \(B\) containing \(E\) such that \begin{align*} m_*(B) = m_*(E) .\end{align*}
Prove that if \(E\subseteq {\mathbb{R}}\) has the property that \begin{align*} m_*(A) = m_*(A\cap E) + m_*(A\cap E^c) \end{align*} for every set \(A\subseteq {\mathbb{R}}\), then there exists a Borel set \(B\subseteq {\mathbb{R}}\) such that \(E = B\setminus N\) with \(m_*(N) = 0\).
Be sure to address the case when \(m_*(E) = \infty\).
Let \((X, \mathcal B, \mu)\) be a measure space with \(\mu(X) = 1\) and \(\{B_n\}_{n=1}^\infty\) be a sequence of \(\mathcal B\)-measurable subsets of \(X\), and \begin{align*} B \mathrel{\vcenter{:}}=\left\{{x\in X {~\mathrel{\Big|}~}x\in B_n \text{ for infinitely many } n}\right\}. \end{align*}
Argue that \(B\) is also a \(\mathcal{B} {\hbox{-}}\)measurable subset of \(X\).
Prove that if \(\sum_{n=1}^\infty \mu(B_n) < \infty\) then \(\mu(B)= 0\).
Prove that if \(\sum_{n=1}^\infty \mu(B_n) = \infty\) and the sequence of set complements \(\left\{{B_n^c}\right\}_{n=1}^\infty\) satisfies \begin{align*} \mu\left(\bigcap_{n=k}^{K} B_{n}^{c}\right)=\prod_{n=k}^{K}\left(1-\mu\left(B_{n}\right)\right) \end{align*} for all positive integers \(k\) and \(K\) with \(k < K\), then \(\mu(B) = 1\).
Hint: Use the fact that \(1 - x ≤ e^{-x}\) for all \(x\).
Let \(\mathcal B\) denote the set of all Borel subsets of \({\mathbb{R}}\) and \(\mu : \mathcal B \to [0, \infty)\) denote a finite Borel measure on \({\mathbb{R}}\).
Prove that if \(\{F_k\}\) is a sequence of Borel sets for which \(F_k \supseteq F_{k+1}\) for all \(k\), then \begin{align*} \lim _{k \rightarrow \infty} \mu\left(F_{k}\right)=\mu\left(\bigcap_{k=1}^{\infty} F_{k}\right) \end{align*}
Suppose \(\mu\) has the property that \(\mu (E) = 0\) for every \(E \in \mathcal B\) with Lebesgue measure \(m(E) = 0\).
Prove that for every \(\epsilon > 0\) there exists \(\delta > 0\) so that if \(E \in \mathcal B\) with \(m(E) < δ\), then \(\mu(E) < ε\).
Let \(E\subset {\mathbb{R}}\) be a Lebesgue measurable set. Show that there is a Borel set \(B \subset E\) such that \(m(E\setminus B) = 0\).
Define
\begin{align*}
E:=\left\{x \in \mathbb{R}:\left|x-\frac{p}{q}\right|
Prove that \(m(E) = 0\).
Let \(f(x) = x^2\) and \(E \subset [0, \infty) \mathrel{\vcenter{:}}={\mathbb{R}}^+\).
Show that \begin{align*} m^*(E) = 0 \iff m^*(f(E)) = 0. \end{align*}
Deduce that the map
\begin{align*} \phi: \mathcal{L}({\mathbb{R}}^+) &\to \mathcal{L}({\mathbb{R}}^+) \\ E &\mapsto f(E) \end{align*} is a bijection from the class of Lebesgue measurable sets of \([0, \infty)\) to itself.
Let \(\mu\) be a measure on a measurable space \((X, \mathcal M)\) and \(f\) a positive measurable function.
Define a measure \(\lambda\) by \begin{align*} \lambda(E):=\int_{E} f ~d \mu, \quad E \in \mathcal{M} \end{align*}
Show that for \(g\) any positive measurable function, \begin{align*} \int_{X} g ~d \lambda=\int_{X} f g ~d \mu \end{align*}
Let \(E \subset {\mathbb{R}}\) be a measurable set such that \begin{align*} \int_{E} x^{2} ~d m=0. \end{align*} Show that \(m(E) = 0\).
Let \((X, \mathcal M, \mu)\) be a measure space and suppose \(\left\{{E_n}\right\} \subset \mathcal M\) satisfies \begin{align*} \lim _{n \rightarrow \infty} \mu\left(X \backslash E_{n}\right)=0. \end{align*}
Define \begin{align*} G \mathrel{\vcenter{:}}=\left\{{x\in X {~\mathrel{\Big|}~}x\in E_n \text{ for only finitely many } n}\right\}. \end{align*}
Show that \(G \in \mathcal M\) and \(\mu(G) = 0\).
Let \(f\) be Lebesgue measurable on \({\mathbb{R}}\) and \(E \subset {\mathbb{R}}\) be measurable such that
\begin{align*}
0
Show that for every \(0 < t < 1\), there exists a measurable set \(E_t \subset E\) such that \begin{align*} \int_{E_{t}} f(x) d x=t A. \end{align*}
Let \((X, \mathcal M, \mu)\) be a measure space. For \(f\in L^1(\mu)\) and \(\lambda > 0\), define \begin{align*} \phi(\lambda)=\mu(\{x \in X | f(x)>\lambda\}) \quad \text { and } \quad \psi(\lambda)=\mu(\{x \in X | f(x)<-\lambda\}) \end{align*}
Show that \(\phi, \psi\) are Borel measurable and \begin{align*} \int_{X}|f| ~d \mu=\int_{0}^{\infty}[\phi(\lambda)+\psi(\lambda)] ~d \lambda \end{align*}
Let \(f: {\mathbb{R}}\to {\mathbb{R}}\) be Lebesgue measurable.
Let \(\mu\) be a finite Borel measure on \({\mathbb{R}}\) and \(E \subset {\mathbb{R}}\) Borel. Prove that the following statements are equivalent:
Let \(f: {\mathbb{R}}\to {\mathbb{R}}\) and suppose \begin{align*} \forall x\in {\mathbb{R}},\quad f(x) \geq \limsup _{y \rightarrow x} f(y) \end{align*} Prove that \(f\) is Borel measurable.
Let \((X, \mathcal M, \mu)\) be a measure space and suppose \(f\) is a measurable function on \(X\). Show that \begin{align*} \lim _{n \rightarrow \infty} \int_{X} f^{n} ~d \mu = \begin{cases} \infty & \text{or} \\ \mu(f^{-1}(1)), \end{cases} \end{align*} and characterize the collection of functions of each type.
Let \(K\) be the set of numbers in \([0, 1]\) whose decimal expansions do not use the digit \(4\).
We use the convention that when a decimal number ends with 4 but all other digits are different from 4, we replace the digit \(4\) with \(399\cdots\). For example, \(0.8754 = 0.8753999\cdots\).
Show that \(K\) is a compact, nowhere dense set without isolated points, and find the Lebesgue measure \(m(K)\).
Let \(0 < \lambda < 1\) and construct a Cantor set \(C_\lambda\) by successively removing middle intervals of length \(\lambda\).
Prove that \(m(C_\lambda) = 0\).
Let \(f, g: [a, b] \to {\mathbb{R}}\) be measurable with \begin{align*} \int_{a}^{b} f(x) ~d x=\int_{a}^{b} g(x) ~d x. \end{align*}
Show that either
Let \(E \subset {\mathbb{R}}\) be measurable with \(m(E) < \infty\). Define \begin{align*} f(x)=m(E \cap(E+x)). \end{align*}
Show that
Hint: \begin{align*} \chi_{E \cap(E+x)}(y)=\chi_{E}(y) \chi_{E}(y-x) \end{align*}
Let \((X, \mathcal{M},\mu)\) be a measure space and let \(E_n \in \mathcal{M}\) be a measurable set for \(n\geq 1\). Let \(f_n \mathrel{\vcenter{:}}=\chi_{E_n}\) be the indicator function of the set \(E\) and show that
\(f_n \overset{n\to\infty}\to 1\) uniformly \(\iff\) there exists \(N\in |NN\) such that \(E_n = X\) for all \(n\geq N\).
\(f_n(x) \overset{n\to\infty}\to 1\) for almost every \(x\) \(\iff\) \begin{align*} \mu \qty{ \bigcap_{n \geq 0} \bigcup_{k \geq n} (X \setminus E_k) } = 0 .\end{align*}
Let \((X, \mathcal{M}, \mu)\) be a finite measure space and let \(\left\{{ f_n}\right\}_{n=1}^{\infty } \subseteq L^1(X, \mu)\). Suppose \(f\in L^1(X, \mu)\) such that \(f_n(x) \overset{n\to \infty }\to f(x)\) for almost every \(x \in X\). Prove that for every \(\varepsilon> 0\) there exists \(M>0\) and a set \(E\subseteq X\) such that \(\mu(E) \leq \varepsilon\) and \({\left\lvert {f_n(x)} \right\rvert}\leq M\) for all \(x\in X\setminus E\) and all \(n\in {\mathbb{N}}\).
Prove that \begin{align*} \left| \frac{d^{n}}{d x^{n}} \frac{\sin x}{x}\right| \leq \frac{1}{n} \end{align*}
for all \(x \neq 0\) and positive integers \(n\).
Hint: Consider \(\displaystyle\int_0^1 \cos(tx) dt\)
Compute the following limit and justify your calculations: \begin{align*} \lim_{n\to\infty} \int_0^n \qty{1 + {x^2 \over n}}^{-(n+1)} \,dx .\end{align*}
Let \(\{f_k\}\) be any sequence of functions in \(L^2([0, 1])\) satisfying \({\left\lVert {f_k} \right\rVert}_2 ≤ M\) for all \(k ∈ {\mathbb{N}}\).
Prove that if \(f_k \to f\) almost everywhere, then \(f ∈ L^2([0, 1])\) with \({\left\lVert {f} \right\rVert}_2 ≤ M\) and \begin{align*} \lim _{k \rightarrow \infty} \int_{0}^{1} f_{k}(x) dx = \int_{0}^{1} f(x) d x \end{align*}
Hint: Try using Fatou’s Lemma to show that \({\left\lVert {f} \right\rVert}_2 ≤ M\) and then try applying Egorov’s Theorem.
Compute the following limit and justify your calculations: \begin{align*} \lim_{n \rightarrow \infty} \int_{1}^{n} \frac{d x}{\left(1+\frac{x}{n}\right)^{n} \sqrt[n]{x}} \end{align*}
Suppose \(f(x)\) and \(xf(x)\) are integrable on \({\mathbb{R}}\). Define \(F\) by \begin{align*} F(t)\mathrel{\vcenter{:}}=\int _{-\infty}^{\infty} f(x) \cos (x t) dx \end{align*} Show that \begin{align*} F'(t)=-\int _{-\infty}^{\infty} x f(x) \sin (x t) dx .\end{align*}
Suppose that
Show that \(\int f_{n} \rightarrow \int f\).
Let \begin{align*} f_{n}(x):=\frac{x}{1+x^{n}}, \quad x \geq 0. \end{align*}
Show that this sequence converges pointwise and find its limit. Is the convergence uniform on \([0, \infty)\)?
Compute \begin{align*} \lim _{n \rightarrow \infty} \int_{0}^{\infty} f_{n}(x) d x \end{align*}
Let \(f\in L^1({\mathbb{R}})\). Show that \begin{align*} \lim _{x \to 0} \int _{{\mathbb{R}}} {\left\lvert {f(y-x)-f(y)} \right\rvert} \, dy = 0 \end{align*}
Compute the following limit: \begin{align*} \lim _{n \rightarrow \infty} \int_{1}^{n} \frac{n e^{-x}}{1+n x^{2}} \, \sin \left(\frac x n\right) \, dx \end{align*}
Let \(f: [1, \infty) \to {\mathbb{R}}\) such that \(f(1) = 1\) and \begin{align*} f^{\prime}(x)= \frac{1} {x^{2}+f(x)^{2}} \end{align*}
Show that the following limit exists and satisfies the equality \begin{align*} \lim _{x \rightarrow \infty} f(x) \leq 1 + \frac \pi 4 \end{align*}
Calculate the following limit, justifying each step of your calculation:
\begin{align*} L \mathrel{\vcenter{:}}=\lim_{n\to \infty} \int_0^n { \cos\qty{x\over n} \over x^2 + \cos\qty{x\over n} }\,dx .\end{align*}
Let \(f_n \in L^2([0, 1])\) for \(n\in {\mathbb{N}}\), and assume that
\({\left\lVert {f_n} \right\rVert}_2 \leq n^{-51 \over 100}\) for all \(n\in {\mathbb{N}}\),
\(hat{f}_n\) is supported in the interval \([2^n, 2^{n+1}]\), so \begin{align*} \widehat{f}_n(\xi) \mathrel{\vcenter{:}}=\int_0^1 f_n(x) e^{2\pi i \xi \cdot x} \,dx= 0 && \text{for } \xi \not\in [2^n, 2^{n+1}] .\end{align*}
Prove that \(\sum_{n\in {\mathbb{N}}} f_n\) converges in the Hilbert space \(L^2([0, 1])\).
Hint: Plancherel’s identity may be helpful.
Let \(f\) be a non-negative measurable function on \([0, 1]\).
Show that \begin{align*} \lim _{p \rightarrow \infty}\left(\int_{[0,1]} f(x)^{p} d x\right)^{\frac{1}{p}}=\|f\|_{\infty}. \end{align*}
Let \(f\in L^2([0, 1])\) and suppose \begin{align*} \int _{[0,1]} f(x) x^{n} d x=0 \text { for all integers } n \geq 0. \end{align*} Show that \(f = 0\) almost everywhere.
Let \(f: {\mathbb{R}}\to {\mathbb{C}}\) be continuous with period 1. Prove that \begin{align*} \lim _{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^{N} f(n \alpha)=\int_{0}^{1} f(t) d t \quad \forall \alpha \in {\mathbb{R}}\setminus{\mathbb{Q}}. \end{align*}
Hint: show this first for the functions \(f(t) = e^{2\pi i k t}\) for \(k\in {\mathbb{Z}}\).
Let \(g\in L^\infty([0, 1])\) Prove that \begin{align*} \int _{[0,1]} f(x) g(x)\, dx = 0 \quad\text{for all continuous } f:[0, 1] \to {\mathbb{R}} \implies g(x) = 0 \text{ almost everywhere. } \end{align*}
Prove that if \(g\in L^1({\mathbb{R}})\) then \begin{align*} \lim_{N\to \infty} \int _{{\left\lvert {x} \right\rvert} \geq N} {\left\lvert {f(x)} \right\rvert} \, dx = 0 ,\end{align*} and demonstrate that it is not necessarily the case that \(f(x) \to 0\) as \({\left\lvert {x} \right\rvert}\to \infty\).
Prove that if \(f\in L^1([1, \infty])\) and is decreasing, then \(\lim_{x\to\infty}f(x) =0\) and in fact \(\lim_{x\to \infty} xf(x) = 0\).
If \(f: [1, \infty) \to [0, \infty)\) is decreasing with \(\lim_{x\to \infty} xf(x) = 0\), does this ensure that \(f\in L^1([1, \infty))\)?
Show that if \(f\) is continuous with compact support on \({\mathbb{R}}\), then \begin{align*} \lim _{y \rightarrow 0} \int_{\mathbb{R}}|f(x-y)-f(x)| d x=0 \end{align*}
Let \(f\in L^1({\mathbb{R}})\) and for each \(h > 0\) let \begin{align*} \mathcal{A}_{h} f(x):=\frac{1}{2 h} \int_{|y| \leq h} f(x-y) d y \end{align*}
Prove that \(\left\|\mathcal{A}_{h} f\right\|_{1} \leq\|f\|_{1}\) for all \(h > 0\).
Prove that \(\mathcal{A}_h f \to f\) in \(L^1({\mathbb{R}})\) as \(h \to 0^+\).
Let \begin{align*} S = {\operatorname{span}}_{\mathbb{C}}\left\{{\chi_{(a, b)} {~\mathrel{\Big|}~}a, b \in {\mathbb{R}}}\right\}, \end{align*} the complex linear span of characteristic functions of intervals of the form \((a, b)\).
Show that for every \(f\in L^1({\mathbb{R}})\), there exists a sequence of functions \(\left\{{f_n}\right\} \subset S\) such that \begin{align*} \lim _{n \rightarrow \infty}\left\|f_{n}-f\right\|_{1}=0 \end{align*}
Define \begin{align*} f(x, y):=\left\{\begin{array}{ll}{\frac{x^{1 / 3}}{(1+x y)^{3 / 2}}} & {\text { if } 0 \leq x \leq y} \\ {0} & {\text { otherwise }}\end{array}\right. \end{align*}
Carefully show that \(f \in L^1({\mathbb{R}}^2)\).
Let \(f\in L^1({\mathbb{R}})\). Show that \begin{align*} \forall\varepsilon > 0 \exists \delta > 0 \text{ such that } \qquad m(E) < \delta \implies \int _{E} |f(x)| \, dx < \varepsilon \end{align*}
Give an example of a continuous \(f\in L^1({\mathbb{R}})\) such that \(f(x) \not\to 0\) as\({\left\lvert {x} \right\rvert} \to \infty\).
Show that if \(f\) is uniformly continuous, then \begin{align*} \lim_{{\left\lvert {x} \right\rvert} \to \infty} f(x) = 0. \end{align*}
Let \(f, g\) be Lebesgue integrable on \({\mathbb{R}}\) and let \(g_n(x) \mathrel{\vcenter{:}}= g(x- n)\). Prove that \begin{align*} \lim_{n\to \infty } {\left\lVert {f + g_n} \right\rVert}_1 = {\left\lVert {f} \right\rVert}_1 + {\left\lVert {g} \right\rVert}_1 .\end{align*}
Let \(f, g\in L^1({\mathbb{R}})\). Argue that \(H(x, y) \mathrel{\vcenter{:}}= f(y) g(x-y)\) defines a function in \(L^1({\mathbb{R}}^2)\) and deduce from this fact that \begin{align*} (f\ast g)(x) \mathrel{\vcenter{:}}=\int_{\mathbb{R}}f(y) g(x-y) \,dy \end{align*} defines a function in \(L^1({\mathbb{R}})\) that satisfies \begin{align*} {\left\lVert {f\ast g} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_1 .\end{align*}
Let \(f\) be a non-negative function on \({\mathbb{R}}^n\) and \(\mathcal A = \{(x, t) ∈ {\mathbb{R}}^n \times {\mathbb{R}}: 0 ≤ t ≤ f (x)\}\).
Prove the validity of the following two statements:
\(f\) is a Lebesgue measurable function on \({\mathbb{R}}^n \iff \mathcal A\) is a Lebesgue measurable subset of \({\mathbb{R}}^{n+1}\)
If \(f\) is a Lebesgue measurable function on \({\mathbb{R}}^n\), then \begin{align*} m(\mathcal{A})=\int _{{\mathbb{R}}^{n}} f(x) d x=\int_{0}^{\infty} m\left(\left\{x \in {\mathbb{R}}^{n}: f(x) \geq t\right\}\right) dt \end{align*}
Let \(f \geq 0\) be a measurable function on \({\mathbb{R}}\). Show that \begin{align*} \int _{{\mathbb{R}}} f = \int _{0}^{\infty} m(\{x: f(x)>t\}) dt \end{align*}
Let \(f, g \in L^1({\mathbb{R}})\) be Borel measurable.
Let \(f, g \in L^1([0, 1])\) and for all \(x\in [0, 1]\) define \begin{align*} F(x) \mathrel{\vcenter{:}}=\int _{0}^{x} f(y) \, dy {\quad \operatorname{and} \quad} G(x)\mathrel{\vcenter{:}}=\int _{0}^{x} g(y) \, dy. \end{align*}
Prove that \begin{align*} \int _{0}^{1} F(x) g(x) \, dx = F(1) G(1) - \int _{0}^{1} f(x) G(x) \, dx \end{align*}
Let \(f: {\mathbb{R}}\times{\mathbb{R}}\to {\mathbb{R}}\) be a measurable function and for \(x\in {\mathbb{R}}\) define the set \begin{align*} E_x \mathrel{\vcenter{:}}=\left\{{ y\in {\mathbb{R}}{~\mathrel{\Big|}~}\mu\qty{ z\in {\mathbb{R}}{~\mathrel{\Big|}~}f(x,z) = f(x, y) } > 0 }\right\} .\end{align*} Show that the following set is a measurable subset of \({\mathbb{R}}\times{\mathbb{R}}\): \begin{align*} E \mathrel{\vcenter{:}}=\bigcup_{x\in {\mathbb{R}}} \left\{{ x }\right\} \times E_x .\end{align*}
Hint: consider the measurable function \(h(x,y,z) \mathrel{\vcenter{:}}= f(x, y) - f(x, z)\).
Show that \begin{align*} L^2([0, 1]) \subseteq L^1([0, 1]) {\quad \operatorname{and} \quad} \ell^1({\mathbb{Z}}) \subseteq \ell^2({\mathbb{Z}}) .\end{align*}
For \(f\in L^1([0, 1])\) define \begin{align*} \widehat{f}(n) \mathrel{\vcenter{:}}=\int _0^1 f(x) e^{-2\pi i n x} \, dx .\end{align*}
Prove that if \(f\in L^1([0, 1])\) and \(\left\{{\widehat{f}(n)}\right\} \in \ell^1({\mathbb{Z}})\) then \begin{align*} S_N f(x) \mathrel{\vcenter{:}}=\sum_{{\left\lvert {n} \right\rvert} \leq N} \widehat{f} (n) e^{2 \pi i n x} .\end{align*} converges uniformly on \([0, 1]\) to a continuous function \(g\) such that \(g = f\) almost everywhere.
Hint: One approach is to argue that if \(f\in L^1([0, 1])\) with \(\left\{{\widehat{f} (n)}\right\} \in \ell^1({\mathbb{Z}})\) then \(f\in L^2([0, 1])\).
Let \(\phi\) be a compactly supported smooth function that vanishes outside of an interval \([-N, N]\) such that \(\int _{{\mathbb{R}}} \phi(x) \, dx = 1\).
For \(f\in L^1({\mathbb{R}})\), define \begin{align*} K_{j}(x) \mathrel{\vcenter{:}}= j \phi(j x), \qquad f \ast K_{j}(x) \mathrel{\vcenter{:}}=\int_{{\mathbb{R}}} f(x-y) K_{j}(y) \, dy \end{align*} and prove the following:
Each \(f\ast K_j\) is smooth and compactly supported.
\begin{align*} \lim _{j \to \infty} {\left\lVert {f * K_{j}-f} \right\rVert}_{1} = 0 \end{align*}
Hint: \begin{align*} \lim _{y \to 0} \int _{{\mathbb{R}}} |f(x-y)-f(x)| dy = 0 \end{align*}
Let \(f, g \in L^2({\mathbb{R}})\). Prove that the formula \begin{align*} h(x) \mathrel{\vcenter{:}}=\int _{-\infty}^{\infty} f(t) g(x-t) \, dt \end{align*} defines a uniformly continuous function \(h\) on \({\mathbb{R}}\).
Let \(f \in L^1({\mathbb{R}})\) and \(g\) be a bounded measurable function on \({\mathbb{R}}\).
Let \(f \in C_c^0({\mathbb{R}}^n)\), and show \begin{align*} \lim _{t \to 0} \int_{{\mathbb{R}}^n} |f(x+t) - f(x)| \, dx = 0 .\end{align*}
Extend the above result to \(f\in L^1({\mathbb{R}}^n)\) and show that \begin{align*} f\in L^1({\mathbb{R}}^n), \quad g\in L^\infty({\mathbb{R}}^n) \quad \implies f \ast g \text{ is bounded and uniformly continuous. } \end{align*}
Let \(\{u_n\}_{n=1}^∞\) be an orthonormal sequence in a Hilbert space \(\mathcal{H}\).
Prove that for every \(x ∈ \mathcal H\) one has \begin{align*} \displaystyle\sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2} \leq\|x\|^{2} \end{align*}
Prove that for any sequence \(\{a_n\}_{n=1}^\infty \in \ell^2({\mathbb{N}})\) there exists an element \(x\in\mathcal H\) such that \begin{align*} a_n = {\left\langle {x},~{u_n} \right\rangle} \text{ for all } n\in {\mathbb{N}} \end{align*} and \begin{align*} {\left\lVert {x} \right\rVert}^2 = \sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2} \end{align*}
Show that \(L^2([0, 1]) ⊆ L^1([0, 1])\) and argue that \(L^2([0, 1])\) in fact forms a dense subset of \(L^1([0, 1])\).
Let \(Λ\) be a continuous linear functional on \(L^1([0, 1])\).
Prove the Riesz Representation Theorem for \(L^1([0, 1])\) by following the steps below:
Hint: You may use, without proof, the Riesz Representation Theorem for \(L^2([0, 1])\).
Without using the Riesz Representation Theorem, compute \begin{align*} \sup \left\{\left|\int_{0}^{1} f(x) e^{x} d x\right| {~\mathrel{\Big|}~}f \in L^{2}([0,1], m),~~ \|f\|_{2} \leq 1\right\} \end{align*}
Let \(\mathcal H\) be a Hilbert space.
Let \(f: [0, 1] \to {\mathbb{R}}\) be continuous. Show that \begin{align*} \sup \left\{\|f g\|_{1} {~\mathrel{\Big|}~}g \in L^{1}[0,1],~~ \|g\|_{1} \leq 1\right\}=\|f\|_{\infty} \end{align*}
Let \(1 \leq p,q \leq \infty\) be conjugate exponents, and show that \begin{align*} f \in L^p({\mathbb{R}}^n) \implies \|f\|_{p} = \sup _{\|g\|_{q}=1}\left|\int f(x) g(x) d x\right| \end{align*}
Let \(C([0, 1])\) denote the space of all continuous real-valued functions on \([0, 1]\).
Prove that \(C([0, 1])\) is complete under the uniform norm \({\left\lVert {f} \right\rVert}_u := \displaystyle\sup_{x\in [0,1]} |f (x)|\).
Prove that \(C([0, 1])\) is not complete under the \(L^1{\hbox{-}}\)norm \({\left\lVert {f} \right\rVert}_1 = \displaystyle\int_0^1 |f (x)| ~dx\).
Show that the space \(C^1([a, b])\) is a Banach space when equipped with the norm \begin{align*} \|f\|:=\sup _{x \in[a, b]}|f(x)|+\sup _{x \in[a, b]}\left|f^{\prime}(x)\right|. \end{align*}
Let \(X\) be a complete metric space and define a norm \begin{align*} \|f\|:=\max \{|f(x)|: x \in X\}. \end{align*}
Show that \((C^0({\mathbb{R}}), {\left\lVert {{\,\cdot\,}} \right\rVert} )\) (the space of continuous functions \(f: X\to {\mathbb{R}}\)) is complete.
Show that if \(x_n\) is a decreasing sequence of positive real numbers such that \(\sum_{n=1}^\infty x_n\) converges, then \begin{align*} \lim_{n\to\infty} n x_n = 0. \end{align*}
Let \(f: {\mathbb{R}}\to {\mathbb{R}}\). Prove that \begin{align*} f(x) \leq \liminf_{y\to x} f(y)~ \text{for each}~ x\in {{\mathbb{R}}} \iff \{ x\in {{\mathbb{R}}} \mathrel{\Big|}f(x) > a \}~\text{is open for all}~ a\in {{\mathbb{R}}} \end{align*}
Recall that a function \(f: {{\mathbb{R}}} \to {{\mathbb{R}}}\) is called lower semi-continuous iff it satisfies either condition in part (a) above.
Prove that if \(\mathcal{F}\) is an y family of lower semi-continuous functions, then \begin{align*} g(x) = \sup\{ f(x) \mathrel{\Big|}f\in \mathcal{F}\} \end{align*} is Borel measurable.
Note that \(\mathcal{F}\) need not be a countable family.
Let \(f\) be a non-negative Lebesgue measurable function on \([1, \infty)\).
Prove that \begin{align*} 1 \leq \qty{ {1 \over b-a} \int_a^b f(x) \,dx }\qty{ {1\over b-a} \int_a^b {1 \over f(x)}\, dx } \end{align*} for any \(1\leq a < b <\infty\).
Prove that if \(f\) satisfies \begin{align*} \int_1^t f(x) \, dx \leq t^2 \log(t) \end{align*} for all \(t\in [1, \infty)\), then \begin{align*} \int_1^\infty {1\over f(x) \,dx} = \infty .\end{align*}
Hint: write \begin{align*} \int_1^\infty {1\over f(x) \, dx} = \sum_{k=0}^\infty \int_{2^k}^{2^{k+1}} {1 \over f(x)}\,dx .\end{align*}
Prove that if \(xf(x) \in L^1({\mathbb{R}})\), then \begin{align*} F(y) \mathrel{\vcenter{:}}=\int f(x) \cos(yx)\, dx \end{align*} defines a \(C^1\) function.
Suppose \(\phi\in L^1({\mathbb{R}})\) with \begin{align*} \int \phi(x) \, dx = \alpha .\end{align*} For each \(\delta > 0\) and \(f\in L^1({\mathbb{R}})\), define \begin{align*} A_\delta f(x) \mathrel{\vcenter{:}}=\int f(x-y) \delta^{-1} \phi\qty{\delta^{-1} y}\, dy .\end{align*}
Prove that for all \(\delta > 0\), \begin{align*} {\left\lVert {A_\delta f} \right\rVert}_1 \leq {\left\lVert {\phi} \right\rVert}_1 {\left\lVert {f} \right\rVert}_1 .\end{align*}
Prove that \begin{align*} A_\delta f \to \alpha f \text{ in } L^1({\mathbb{R}}) {\quad \operatorname{as} \quad} \delta\to 0^+ .\end{align*}
Hint: you may use without proof the fact that for all \(f\in L^1({\mathbb{R}})\), \begin{align*} \lim_{y\to 0} \int_{\mathbb{R}}{\left\lvert {f(x-y) - f(x)} \right\rvert}\, dx = 0 .\end{align*}