# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### A sufficient condition for almost everywhere equality

Let $f,g:(0,\infty)\to \mathrm{R}$ be monotone decreasing functions. Show that if $m(\{x:f(x)>a\})=m(\{x;g(x)>a\}),\; \forall a\in \mathrm{R}$ where $m$ denotes Lebesgue measure, ...
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### limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
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### Inclusions relating standard norms, in measury theory

I know that for finite measure space $(X, \mathcal A ,\mu )$ and $1\leq p< q<\infty$ , the inclusion $\mathcal L^q\subseteq \mathcal L^p\subseteq\mathcal L^1$ holds true (applying Holder's ...
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### On the equality of two sets (a doubt from Probability with Martingales).

Let $(S, \Sigma, \mu)$ be $([0,1], \mathcal{B}[0,1], Leb)$. Let $\epsilon(k)$ be a sequence of strictly positive numbers s.t. $\epsilon(k) \downarrow 0$. Let $V = Q \cap [0,1],$ the set of rationals ...
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### lebesgue measure basic exercise

I have a basic question about a Lebesgue measure exercise that I am not sure how to solve. (I apologize if this is a simple question, I am new with this subject). Compute the Lebesgue measure of $X$ ...
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### for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
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### Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1]$). I know about ...
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### Characterization of the space of integrable functions stable under multiplication

Let $Ω = (Ω,Σ_Ω,μ)$ be a measure space and let $L(Ω)$ be the space of integrable functions on $Ω$. For $f ∈ L(Ω)$, set $L_f(Ω) = \{φ ∈ L(Ω);~f·φ ∈ L(Ω)\}$. Has the space of all integrable functions ...
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### If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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### How can I prove that $f$ and $g$ are measurable functions [closed]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
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### Are random variables in a tail σ-algebra in the same probability space?

Let $X_1, X_2, ...$ be random variables. Define $\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, ...)$ and $\mathscr{T} = \bigcap_{n} \mathscr{T}_n$, the tail σ-algebra of $X_1, X_2, ...$. When defining a ...
$\Omega_1$ and $\Omega_2$ are countable sets. With $\mathcal P(\cdot)$ we denote a power set of a set. We need to proof that: \mathcal P(\Omega_1)\otimes \mathcal P(\Omega_2)=\mathcal P(\Omega_1 \...
Let $A$ be a Lebesgue measurable set in $\mathbb {R}$ with a positive measure. Then, show that for any positive real number $r$, there is an open interval $I$ such that \$\operatorname{m} (A\cap I)&...