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The question has already been answered satisfactorily, but let me give a reference to a more general statement. If $X$ is a topological space with a finite measure $\mu$ defined on the $\sigma$-algebra of sets having the property of Baire, and if $\mu(E)=0$ iff $E$ is meager, then $(X,\mu)$ is said to be a category measure space. So your question, after ...

5

It follows easily from the following result. Lemma. For any finite Borel measure $\mu$ on $[0,1]$ (possibly $\mu=0$) and $\varepsilon>0$, there exists $1/3<a<1/2<b<2/3$ such that $[0,a]\cup[b,1]$ has measure at least $(1-\varepsilon)\mu([0,1])$. Indeed, you can then iterate the argument with $\varepsilon$ going to zero fast enough that the ...

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I believe I found an answer. Note that in a similar way that we constructed $P_\mu,$ we may define $P_x$ as the unique Borel probability on $M^{\mathbb N}$, such that given $A_0,\ldots,A_n \in M,$ then P_{x}\left(\{\omega_n\}_{n\in\mathbb N} \in M^{\mathbb N}; x_i\in A_i, \ \forall \ i\in\{0,1,\ldots,n\}\right) = \int_{A_0}\int_{A_1} \ldots \int_{A_{n-1}} ...

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As note 'added in proof': the above answers in the positive the point I was trying to make, i.e. that in the presence of a gap at some $r>0$, the measure $\mu$ cannot be atomless. Indeed, suppose that $\mu$ has a gap at $r$. Let $\alpha=\sup\{\mu(A):A\in \cal C\}$ and $\beta=\inf\{\mu(A):A\in \cal M, \mu(A)\ge r\}$. If $\mu$ were atomless, we know that ...

3

Arbitrary metric spaces, no. A classic text discussing such "derivation" is: Hayes, C. A.; Pauc, C. Y., Derivation and martingales, Ergebnisse der Mathematik und ihrer Grenzgebiete. 49. Berlin-Heidelberg-New York: Springer-Verlag. VII, 203 p. (1970). ZBL0192.40604. I think you get trouble even in $\mathbb R^2$ if you define your metric so that the $... 3 Let$F$be a closed subset of$[0,1]$(say) with empty interior and$\mu(F)>0$(where$\mu$is Lebesgue measure), for example given by a fat Cantor set. Clearly,$1_F$is Baire class$1$. I claim that$1_F$cannot be almost everywhere equal to an almost everywhere continuous function. Indeed, assume that$f = 1_F$almost everywhere, say$f(x) = 0$if$x ...

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My question has been answered in this MathStackexchange post: https://math.stackexchange.com/questions/4033589/sobolev-space-with-negative-index For every $k<0$, there exist a measure $\mu_k$ which is singular with respect to Lebesgue measure, and such that $\mu_k\in W^{k,2}(\mathbb R^n)$. So $W^{k,2}(\mathbb R^n)$ does not embed into the space of ...

2

The answer, in general, is negative. That is, there exist continuous maps $T:X \to X$ of the type you describe and Borel probability measures $\mu$ such that, for small $\epsilon$, the set $B_\epsilon(\mu)$ is empty. For instance, this happens whenever $T$ is uniquely ergodic [1] (The canonical example of that is an irrational rotation on the circle.) ...

1

This is a comment which I hope will be of interest to you, but it will be too long for this format. The type of conditions that you mention were analysed in some detail by Walter Schachermayer in the 70´s in the context of functional analysis. I am not sure if the term measure space has a universally accepted meaning but the basic setting he used was that ...

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$\newcommand{\ep}{\varepsilon}$Let \begin{equation*} c:=\inf f,\quad d:=\sup f,\quad I:=[a,b],\quad F:=D_f, \end{equation*} so that \begin{equation*} F(y)=\mu(\{t\in I\colon f(t)\le y \})\quad \forall y\in[c,d]. \end{equation*} Also introduce the set \begin{equation*} E_t:=\{y\in[c,d]\colon F(y)\ge t\}\quad \forall t\in[0,b-a]. \end{equation*} ...

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It is not true in general that the function $p$ is concave. For instance, let $\Omega$ be the conical hull of the ball of radius $1$ centered at the point $(2,0,\dots,0)\in\mathbb R^n$. Then $p(x)=c_nx^{n-1}$ for some real $c_n>0$ depending only on $n$ and for all real $x\ge0$. So, $p$ is not concave even on the interval $[0,\infty)$ if $n\ge3$. However, ...

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Since you refer to transportation metrics, it seems to be fair to assume that your probability spaces are Lebesgue (=standard). Then it is the same to talk either about "lifting" your measure $\mu$ to $\gamma$ or about the family of the conditional measures of the projection $\gamma\to\mu$ (or, which is also the same, about the corresponding ...

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