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Let $\Omega$ be a set $\mathcal A$ be a $\sigma$-algebra on $\Omega$ $\mu:\mathcal A\to\mathbb R$ be $\sigma$-additive If $\mathcal S\subseteq2^\Omega$ with $\emptyset\in\mathcal S$, then $$\opera …

Let $(\Omega,\mathcal A)$ be a measurable space $E$ be a $\mathbb R$-Banach space $\mu:\mathcal A\to E$ with $\mu(\emptyset)=0$ and $$\mu\left(\biguplus_{n\in\mathbb N}A_n\right)=\sum_{n\in\mathbb N …

Let $(\Omega_i,\mathcal A_i)$ be a measureable space $\mathcal M_i\subseteq2^{\Omega_i}$ be a $\pi$-system with $\Omega_i\in\mathcal M_i$ and $\sigma(\mathcal M_i)=\mathcal A_i$ $\mathcal E(M_1\time …

Let $\Omega$ be a set $\mathcal S\subseteq2^\Omega$ be a set with $\emptyset\in\mathcal S$ $\mathcal S_{\text{loc}}:=\left\{A\subseteq\Omega:A\cap S\in\mathcal S\text{ for all }S\in\mathcal S\right\ …

Let $E$ be a Hausdorff space and $\mu$ be a tight$^1$ finite measure on $E$. Is it possible to show that there is a closed separable $E_0\subseteq E$ such that $\mu(E_0)=\mu(E)$? If not, I'm also int …

Let $E$ be a normed $\mathbb R$-vector space. If $x:[0,\infty)\to E$ is càdlàg, let $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t\ge0$$ ($x(0-):=0$) and $$\Delta x(t):=x(t)-x(t-)\;\;\;\text{for }t\ge0 …

If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, …

Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$. I'm searching for a Radon-Nikodým-like theo …

Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $p\ge1$, $E_i$ be a $\mathbb R$-Banach space, $U_1\subseteq E_1$ be open and $f:U_1\to L$ be Fréchet differentiable at $x\in U_1$, whe …

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$, $\varphi_i:E'\to E$ be bijective a …

Let $(E,\mathcal E,\lambda)$ be a probability space and $\lambda$ be a measurable map on $(E,\mathcal E)$ with $\lambda\circ\tau^{-1}=\lambda$. I would like to show that $\tau$ is $\lambda$-ergodi …

Let $E$ be a metric space, $\mathcal M(E)$ denote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$, $(\mu_t)_{t\in I}$ be a net …

Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1 …

Let $(X_t)_{t\ge0}$ be a càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$ and $B\in\mathcal B([0,\infty)\times\mathbb R)$. How can we s …

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(X_t)_{t\in[0,\:\infty]}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$; $\tau$ be an $[0,\infty]$-valued random …