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No, by Bochner's Theorem the Fourier transform of a measure is positive definite and hence bounded.

You can always formulate this problem as an optimization/feasibility problem and look for solutions. The conditions on the marginals gives two linear constraints: $$P_1\pi_\epsilon = \mu,\quad P_2\pi_ …

Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out. Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure …

The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit $$ \lim_{R\to\infty}|\nu-\mu|(B_R) $$ exists. Considering t …

Complementing the previous answer and comment: These metric is used in the context of shape optimization problems (in the case of $X$ a subset of $\mathbb{R}^d$ with the Lebesgue measure). In "Shapes …

Here is a question about decomposition of measures in singular parts and in positive and negative parts. $\newcommand{\RR}{\mathbb{R}}$ Let $\Omega_{1/2}$ be compact subsets of $\RR^d$ equipped signe …

I think, you are looking for a "monotone class argument", see the wikipedia entry for the monotone class theorem. In a nutshell, prove that the property holds for indicators and is preserved under fin …

Whether or not the limit is singular (e.g. with respect to the Lebesgue measure), there are several notions of convergence for measures which can reflect this. Probably the most simple one is weak con …

I quote "Functional Analysis for Probability and Stochastic Processes: An Introduction" by Adam Bobrowski, introduction to Chapter 3 "Conditional expectation": $\newcommand{\cF}{\mathcal{F}}\newcomman …

If you work on compact domains instead of $\RR$ (or if you assume that all marginals have compact support), you can use couplings from regularized optimal transport: For $n=1$ (i.e. a coupling of just …

For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space …

Edit: Changed from "Hausdorff" to "metric" spaces. Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, this is the dual spa …

One thing that is straightforward would be to extend $F$ to the full space $\mathcal{M}$ of signed measures by setting $F(\mu) = -\infty$ if $\mu$ is not a probability measure. This extension would pr …

You may look at the Kantorovich-Rubinstein norm mentioned in this answer. It also goes under the name "flat norm" but I don't know a reference for this. Standard references for these things are the "M …

I thought, I could turn the comments into an answer… The approach by couplings does not work without modifications and the reason is that couplings do not exist if the measures have different total m …