Articles of measure theory

How do I go about proving da db/a^(-2) is a left Haar measure on the affine group?

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 $ […]

Question on Functions of Bounded Variation

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 $x\in [a,b],$ […]

Does $G_{\delta}+q$ sets cover $\Bbb{R}$ a.e

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 […]

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

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. […]

Is $\sup_{E\ \text{of finite measure}}\mu(E)<+\infty$ equivalent to $\mu(X)<+\infty$ in $(X,\Sigma,\mu)$?

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 $\mu(X)=+\infty$?

Nonmeasureable subset of ${\mathbb{R}}^2$ such that no three points are collinear?

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 […]

H. Steinhaus Theorem simple proof

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}\; […]

Generalization of counting measure is a measure

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 […]

Independence of Random Variables By Guessing

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 […]

Question (2) on Uniform Integrability

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 […]