Let $A\subset R$. Then, $|A|=\lim_{t\to \infty}|A\cap (-t,t)|\tag 1$ $|A|:=\inf\{\sum_j l(I_j): A\supset \cup I_j\},$ where $I_j$'s are open intervals in $\mathbb R$ and $l(I_j)$ is the length of the interval $I_j$, and infimum is taken over all sequences of open intervals whose union contains $A...

This question comes from Bass, Ex 4.15. Given a finite measure space $(X, \mathcal{A}, \mu)$, we can define an outer measure $\mu^*$ as $$ \mu^*(A) := \inf\{ \mu(B) : A \subset B , B \in \mathcal{A} \} \:. $$ (One can check that this is an outer measure and that $\mu^*$ agrees with $\mu$ on $\m...

Let $\{E_n\}$ be an increasing sequence of subsets of $\mathbb{R}^n$, measurable or not. Then $$m^* \bigg( \bigcup_{n=1}^{\infty}E_n \bigg)=\lim_{n\rightarrow\infty}m^*E_n$$

Elsewhere, among a group of high school math teachers, I encountered a discussion of the term 'cancel'. Most (>20) people in the discussion had very strong feelings about why the term should be eliminated. I searched among posts on this stack to find the word common used and its meaning seems wel...

Find a power series solution about the regular singular point $x_0=0$ of the differential equation: $$x^2\frac{d^2y}{dy^2}-x\frac{dy}{dx}+(x^2+1)y=0.$$ My Attempt: We assume the power series solution to be of the form $y=\sum_{n=0}^{\infty}c_nx^{n+r},$ such that $c_0\neq 0.$ Now, $y'=\sum_{n=0}^{...

Let me consider the map $$T:\Bbb{C}\setminus \{-i\}\rightarrow \Bbb{C}, ~z\mapsto \frac{z-i}{-iz+1}$$ I want to look at it as a map from the upper half plane to the unit disk, because there I know that it is an isometry if I endow the upper half plane with the Riemannian metrig $g_\Bbb{H}= \frac...

Let $n>-1$ be an integer and $f(n)$ be the integer part of $n^{\frac{5}{4}}$. And to avoid confusion define $f(0)=0,f(1)=1$ and all $f(n)$ as the integer part of the positive real values for $n^{\frac{5}{4}}$. ( no complex or negative branches for the power ) Conjecture : Every positive integer c...