Articles of measure theory

Prove random vector with covariance matrix $\Sigma$ has non-degenerate distribution iff $\Sigma$ is positive definite

Let $X$ denote a $d$-dimensional random vector with covariance matrix $\Sigma$ satisfying $|\Sigma| < \infty$. Prove $X$ has non-degenerate distribution iff $\Sigma$ is positive definite. Attempt at proof: Let $X$ have a non-degenerate distribution. Then $$\mathrm{Pr}(a_1 X_1 + a_2 X_2+\cdots+a_n X_n=c) \neq 1.$$ From here I’m not sure how to proceed. However, I believe it […]

About a solution of Measure Theory and Integration

The problem is from Folland’s book of Measure Theory and Integration. In this problem, $(X,\mathcal{M}, \mu)$ is the measure space and $L^+$ is the space of measurable functions $f:X\to[0,\infty]$. The problem and the solution are showed below. I have just a little question about this solution. In this context, when we are talking about simple […]

Basis in $L^2()$

Consider the space $L^2([-\pi,\pi])$. Show that the functions $f_0(x)=1,f_1(x)=x,f_2(x)=x^2,\ldots $ form a basis. The functions are linearly independent (no linear combination adds up to zero). But for them to form a basis, doesn’t their linear combination have to hit all functions in $L^2([-\pi,\pi])$? Surely not all functions in $L^2([-\pi,\pi])$ can be written as a polynomial. […]

Absolute continuous family of measures

Consider the following family of measures on $(\mathbb R,\mathcal B(\mathbb R))$: $$ K_x(A) = \begin{cases} \int\limits_A \frac{1}{|x|\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy&,\text{ if }x\neq 0, \\ I_{A}(0)&,\text{ if }x = 0. \end{cases} $$ where $I_A(t)$ is an indicator (characteristic) function of the set $A$. I wonder if it is possible to find a measure $\mu$ such that $\mu>>K_x$ for […]

Proving a sufficient and necessary condition for $f:\, X\to\mathbb{R}\cup\{\pm\infty\}$ to be measurable

I saw the following question: Denote $\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}$, the open sets containing $x\in\mathbb{R}$ are the open sets in $\mathbb{R}$ containing $x$. The open sets containing $\pm\infty$ are the sets of the form $V\cup\{\infty\}$ or $V\cup\{-\infty\}$ accordingly. Let $(X,S)$ be a measurable space. Prove that $f:\, X\to\overline{\mathbb{R}}$ is measurable iff the following conditions hold: $f^{-1}(\{\infty\}),f^{-1}(\{-\infty\})\in S$ $f$ […]

Countably generated versus being generated by a countable partition

(1) Apparently a general term of a sigma-field generated by a countable partition can be written down. For example, if $\mathcal{B} = \sigma(B_n,n\ge 1)$ and $\{B_n\}_{n\ge1}$ is a partition of the ground set $\Omega$, then a general element of $\mathcal{B}$ is of the form $\cup_{n \in I} B_n$ for some $I \subset \mathbb{N}$. (2) Apparently, […]

Lebesgue-integrable function

I have to decice if the following function is Lebesgue-integrable on $[0,1]$: $$g(x)=\frac{1}x\cos\left(\frac{1}x\right) $$ where $x\in[0,1]$.

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is taken over the real line in both directions, and the weighting is roughly exponential. Hilbert Polynomials are complete using a different weighting, while Laguerre polynomials use a similar exponential weighting, but only on the positive real line. What […]

A continuous function defined by Lebesgue measure

Let $A,B\in\mathcal A_{\Bbb R}^*$ given with $\overline{\lambda}(A)<\infty$ and $\overline{\lambda}(B)<\infty$. Lets define $\; \overline{\lambda}_{A,B}:\Bbb R\to\Bbb R$ as follows: $$\overline{\lambda}_{A,B}(x)=\overline{\lambda}(A\cap(B+x))$$ where $B+x=\{b+x:b\in B\}$. So what I want to prove is that $\overline{\lambda}_{A,B}$ is continuous. Proof: Let $c\in\Bbb R$ fixed and let $x_n=c-\frac{1}{n}\;\forall n\in\Bbb N$. Clearly $x_n\in\Bbb R\;\forall n\in\Bbb N$. Then, let $B_n=B+x_n\;\forall n\in\Bbb N\Rightarrow\ B_n=\{b+c-\frac{1}{n}:b\in B\}\;\forall n\in\Bbb […]

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$ I am […]