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Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence. Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the …

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $$ \su …

No I do not think this is always possible. Take for example a subsequence of $\mathbb{N}$, call it $\{ m_k \}$ and consider the random variables \begin{equation} X_i = \chi_{(0,1/2)},\quad m_k \leq i …

Suppose that $\mu$ is a signed finite Borel measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that \begin{equation …

The so called integral Minkowski inequality claims that for suitable positive functions $f$ defined on the product of two measure spaces $ (X\times Y, \mu \times \nu) $ and $p\geq 1$ we have that $$ \ …

I think it goes like this. \begin{align*} P \big(\overline{P(\overline{f} \circ \varphi_\lambda}) \big) (w) & = \int_\Omega \overline{P(\overline{f} \circ \varphi_\lambda}) (z) \overline{k_w(z)} dV(z) …