Let $ G $ be the affine group, in other words $ G := \mathbb{R}\times \mathbb{R}\backslash\{0\} $ with binary operation defined by $ (b,a)\cdot(x,s) = (ax+b,as) $. Now, $ G $ is a locally compact group and thus has a left (resp. right) Haar measure. In particular I know that $ d\mu = a^{-2}da\,db $ […]

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) – f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions of $[a,b]$ My question is: If $f \in BV[a,b],$ show that $|f(x)|\leq |f(a)| + V_f[a,b] \ \ $ for all $x\in [a,b],$ […]

Let $G_{\delta}$ be countable intersections of given open sets with positive Lebesgue measure on $[a,b]$. My question is that if $G_{\delta}+q$ covers $\Bbb{R}$ a.e, i.e. is $$ \bigcup_{q \in \mathbb{Q}}(q+G_{\delta})=\Bbb{R}-N $$ true? ($N$ is of Lebesgue measure zero). $G_{\delta}$ must be uncountable for it has positive Lebesgue measure. But it may has empty interior. I […]

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to $L_2\left(\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}\right)$. It seems to me that desired isometric isomorphism is $$ i:\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}\to L_2\left(\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}\right):f\mapsto(x_\nu\mapsto f(\nu)(x_\nu))\qquad x_\nu\in X_\nu,\quad\nu\in\Lambda $$ Here is my proof: $$ \Vert i(f)\Vert^2=\int\limits_{\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}}|i(f)(x)|^2 d\left(\sqcup\{\mu_\nu:\nu\in\Lambda\}\right)(x)= $$ $$ \sum\limits_{\nu\in\Lambda}\quad\int\limits_{X_\nu}|i(f)(x_\nu)|^2d\mu_\nu(x_\nu)= \sum\limits_{\nu\in\Lambda}\quad\int\limits_{X_\nu}|f(\nu)(x_\nu)|^2d\mu_\nu(x_\nu)= $$ $$ \sum\limits_{\nu\in\Lambda}\Vert f(\nu)\Vert^2=\Vert f\Vert^2.$$ Hence $i$ is an isometry. […]

This relates to a previous question I posted. Let $(X,\Sigma,\mu)$ be a measure space with $\mu(X)<+\infty$. Then $\sup_{E\in{\mathcal A}_{\infty}}\mu(E)<+\infty$, where $ {\mathcal A}_{\infty}=\{A\in\Sigma:\mu(A)<+\infty\}. $ My question is: Is there a measure space such that $\sup_{E\in{\mathcal A}_{\infty}}\mu(E)<+\infty$ but $\mu(X)=+\infty$?

I’m exploring the properties of sets in the plane that do not contain a set of three collinear points. In particular, I’m interested in the “largest” they can be. Things I know so far: Assuming the axiom of choice, one can use Zorn’s lemma to show that there are maximal subsets of ${\mathbb{R}}^2$ with this […]

Let $A,B\in\mathcal A_{\Bbb R}^*$ be given with $\overline{\lambda}(A)<\infty$ and $\overline{\lambda}(B)<\infty$. Let $\overline{\lambda}(A)>0$ I want to prove that: $$\exists\ \delta>0\;\text{s.t.}\;\;\text{if}\;\;|x|<\delta\ \Rightarrow\ \overline{\lambda}(A\cap (A+x))>0\;\;\text{and}\;\;(-\delta,\delta)\subset \{a-a’:a,a’\in A\}=:A-A$$ Proof: I’ve already show that $\overline{\lambda}_{A,B}(x)=\overline{\lambda}(A\cap(B+x))$ is continuous $\forall x\in\Bbb R$ here. So, since $\overline{\lambda}_{A,A}$ is continuous and $\overline{\lambda}_{A,A}(0)=\overline{\lambda}(A)>0$ $$\Rightarrow\ \exists\ \delta>0\;\; \text{s.t.}\;\; \overline{\lambda}_{A,A}(x)>0\;\; \text{if}\;\; |x|<\delta$$ where $\overline{\lambda}_{A,A}(x)=\overline{\lambda}(A\cap (A+x))$ $$\Rightarrow\ \text{if}\; […]

Let $X, \mathcal{P}(X) = M$ be a $\sigma$-algebra. Let $f: X \rightarrow [0,\infty]$ be a function. Define $\mu : M \rightarrow [0,\infty]$ by $\mu(E) = \sum_{x \in E} f(x)$, which Folland defines as $$\sup_{\substack{F \subset E \\ F \text{ finite}}} \sum_{x \in F} f(x).$$ Prove that $\mu$ is a measure. It’s easy to show that […]

Our lecturer said that if two random variables are independent, it should usually be “obvious” from their joined density. On the following examples, he then indeed proceeded to prove independence by guessing what the marginal densities would be. I am not sure as to why that works, here’s his reasoning (I tried following it as […]

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W) = 1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that $\forall w \in W$ $\ x \mapsto f(x,w)$ is continuous; $\forall x \in X$ $\ w \mapsto f(x,w)$ is measurable. Assume: (1) $\forall x \in […]

Intereting Posts

Have I found an example of norm-Euclidean failure in $\mathbb Z $?
Graph-theory exercise
Find the common divisors of $a_{1986}$ and $a_{6891}$
What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?
Determine whether $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in $ is uniformly convergent
Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice
'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$
What happens to the frequency-spectrum if the signal gets reset periodically?
Space Sobolev $W^{m,p}$ complete
$n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$
On terms “Orientation” & “Oriented” in different mathematical areas?
Is there a difference between writing $f: X\rightarrow Y$ and writing $f:X\mapsto Y$?
Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$
Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$
floating-point operations do not satisfy the well-known laws for arithmetic operations