Articles of analysis

Proof involving norm of an integral

I am totally stuck and have no idea whatsoever on how to prove the following inequality (by the way this is a problem from an undergraduate book in multivariable advanced calculus at Junior/Senior level ): Let $g=\left ( g_{1},g_{2},…,g_{n} \right ): \left [ a,b \right ]\rightarrow \mathbb{R}^{n}$ is a continuous function, then we define: $\int_{a}^{b}g\left […]

$X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, show that $d(a,b) = d(K,F)$ for some $a$, $b$

Some caveats: Let $K$ be non-empty and compact, $F$ be non-empty and closed, $X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, where $d(x,y)$ is one of the metrics $d_1$, $d_2$, or $d_\infty$ on $\mathbb{R}^n$. Show that $d(a,b) = d(K,F)$ for some $a\in K$, $b\in F$. It seems […]

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to prove this, for each $n \in \mathbb{Z}$ let $E_n=E \cap (n,n)$. How to show that the regularity theorem for $E_n$ can […]

Riemann-Stieltjes integral of unbounded function

In many theorems about the Riemann-Stieltjes integral they required the hypothesis of $f$ to be bounded (for example: Suppose that $f$ is bounded in $[a,b]$, $f$ has only finitely many points of discontinuity in $I=[a,b]$, and that the monotonically increasing function $\alpha$ is continuous at each point of discontinuity of $f$, then $f$ is Riemann-Stieltjes […]

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, …, $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p’}(E)$, where $p’$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$ This is from a past qual. Not really sure […]

Prove that $\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$

Suppose that the measurable sets $A_1,A_2,…$ are “almost disjoint” in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ Conversely, suppose that the measurable sets $A_1,A_2,…$ satisfy $$\mu\left(\cup_{k=1}^\infty A_k\right) = \sum_{k=1}^\infty\mu(A_k)<\infty$$ Prove that the sets are almost disjoint. Here $\mu(A)$ denotes the Lebesgue measure of $A$. I know that if […]

Order of $\frac{f}{g}$

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Prove that if $f$ and $g$ are entire functions of finite order $\rho$, and $\frac{f}{g}$ is entire,then $\frac{f}{g}$ is of order $\leq […]

Should a metric always map into $\mathbf{R}$?

Typically you see the definition of a metric as a function which maps $X\times X\to\mathbf{R},$ but does this always have to be the case? Motivating example: When you complete $\mathbf{Q}$ with the Archimedean metric you think of it as mapping into $\mathbf{R},$ but if you were to choose the $p$-adic metric instead it can only […]

if $A$ has Lebesgue outer measure $0$ then so does $B=\left\{x^2: x\in A \right\}$

Let $A$ be a subset of $\mathbb{R}$ such that $\mu^*(A)=0$, where $\mu^*$ is the Lebesgue outer measure. Prove that if $B=\left\{x^2: x\in A \right\}$, then $\mu^*(B)=0$. Recall that the Lebesgue outer measure of a subset $A$ of $\mathbb{R}$ is defined as $$ \mu^*(A)=\inf \left\{\sum_{i \geq 1} (b_i-a_i): (a_i, b_i) \subseteq \mathbb{R}, A\subseteq \cup_{i\geq 1} (a_i, […]

Proof that rational sequence converges to irrational number

Let $a>0$ be a real number and consider the sequence $x_{n+1}=(x_n^2+a)/2x_n$. I have already shown that this sequence is monotonic decreasing and thus convergent, now I have to show that $(\lim x_n)^2 = a$ and thus exhibit the existence of a positive square root of $a$. (because we took $x_1 > 0$