Articles of real analysis

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{h}_{1}\left(\left.\theta_{1}\right|\theta_{2}\right)\cdots\hat{h}_{K}\left(\left.\theta_{K}\right|\alpha\right)d\theta_{1}\cdots d\theta_{K} $$ Here, the density function $\hat{f}\left(\left.y\right|\theta_{1}\right)$ depends on parameter $\theta_{1}$ which is unknown and is governed by another density function, $\hat{h}_{1}\left(\left.\theta_{1}\right|\theta_{2}\right)$ with hyper-parameter $\theta_{2}$ which could again be governed by another density $\hat{h}_{2}\left(\left.\theta_{2}\right|\theta_{3}\right)$ with hyper-parameter $\theta_{3}$ and so no until we have density […]

How to solve the differential equation $y' = \frac{x+y}{x-y}$?

Solve the following differential equation: $y’ = \frac{x+y}{x-y}$ Someone please help me start this problem. This does not look like a regular first-order differential equation in the form $y’ + 2xy = x$. Thank you.

Lipschitz continuity of atomless measures

Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be Lipschitz continuous in the first argument, i.e. there exists $L \in \mathbb{R}_{>0}$ such that for all $x,y \in \mathbb{R}^n$ we have $\sup_{w \in \mathbb{R}^m} | f(x,w)-f(y,w) | \leq L |x-y|$. Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$ be an atomless probability measure. Assume $f$ is measurable. I am wondering […]

$E \subset \mathbb R$ is an Interval $\iff E$ Is connected

My text gives the definition that $E$ is disconnected if there exist disjoint open sets $A, B$ such that: $A \cap E$, $B \cap E$ are nonempty. $(A \cap E) \cup (B \cap E) = E$. Then for the $(\implies)$ direction, the text offers the following as proof: Suppose $E$ is an interval and $E$ […]

Closure of a set of real-valued functions…

Let $\mathcal F(\mathbb R)$ be set all of real valued function on $\mathbb{R}$ and $S\subset \mathcal F(\mathbb R)$ such that $f\in S$ if only if there is an interval $I$ and a polynomial $p\in \mathbb{R}[x]$ such that $$f(x)=p(x)$$ for all $x\in I$. Now consider the set $\bar{S}$, where $f\in \bar{S}$ if only if there is […]

Equality of limits on $\varepsilon – \delta$ proof

Let $f(x,y)$ be a real-valued function defined on an open set $S$ containing the origin. Prove the following by $\varepsilon – \delta$ definition: If there exists: $$\lim_{(x,y)\to (0,0)} f(x,y)=L,$$ and there exists: $$\lim_{x \to 0}\lim_{y \to 0} f(x,y)=L_{12},$$ then $L=L_{12}$. I’m trying to work out something like |$f(x,y)-L$|<$\varepsilon$, and maybe apply the triangular inequality, but […]

Chain rule for second derivative

If $f:A\subset\mathbb{R^n}\to \mathbb{R}^m$ and $g:B\subset\mathbb{R}^m \to \mathbb{R}^p$ are twice differentiable and $f(A)\subset B$, then for $x_0\in A$, $x,y\in \mathbb{R}^n$, show that: $$D^2(g\circ f(x_0))(x,y)=D^2g(f(x_0))(Df(x_0)\cdot x, Df(x_0)\cdot y)+Dg(f(x_0)) \cdot D^2f(x_0)(x,y) $$ I know that I should apply chain rule twice and get: $$D^2(g\circ f (x_0))(x,y)=D^2g(f(x_0))Df(x_0)(x,y)+Dg(f(x_0))D^2f(x_0)(x,y)$$ But I’m a little confused with the notation and the vector product, […]

The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$

I am looking for the solution of $$\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{ M \sin rx}\right|$$ where $M < N$ are integers and $x \in \mathbb{R}^+$. For $M = 4, N = 6$, $f_{r,M}(x) =\left|\frac{\sin rMx}{ M \sin rx}\right|$ is plotted for different values of $r$ in the figure below. The maximum is plotted […]

theorem of regularity of Lebesgue measurable set

I was reading a proof regarding the condition for Lesbesgue measurable set. Specifically it is the Theorem 2.24 and the proof of the theorem here: In the theorem, it says a set A is lebesgue measurable if and only if there is an open set G where $A\subset G$ such that $\mu^{*}(G\setminus A) \le […]

Uncountability of the Cantor Set

Let $x=\left(0.a_1a_2…\right)_3 \in \mathcal{C}$, where $\mathcal{C}$ denotes the Cantor set and the $a_i$’s are either $0$ or $2$. Let $f:\mathcal{C}\rightarrow \left[0,1\right]$ such that $f(x)=\left(0.(a_1/2)(a_2/2)…\right)_2$. I ultimately want to show that $\mathcal{C}$ is uncountable. But before I must show that $f$ is continuous and surjective. This is my attempt for the surjective part: Let $y\in [0,1]$. […]