I'm reading a paper about metric measure spaces but I have some doubts on how those two things can hold: let $(X,d)$ be a Polish space and $\mathfrak{m}$ a non-negative Radon measure on $X$.
- Let $C\subset D\subset X\times X$ with $C$ open subset, $\pi$ a probability measure on $X\times X$ such that $\pi(D)=1$ and assume that $\mathrm{supp}\,\mathfrak{m}=X$. Then is it true that there exist countably many Borel sets $A_i,B_i\subset X$ such that $$\pi\bigg(\bigcup_{i\in\mathbb{N}}A_i\times B_i\bigg)=1,\qquad\bigcup_{i\in\mathbb{N}}A_i\times B_i\subset D?$$I know that the set $C$ is open and that I can approximate it with countably many balls, but this works for $C$ and not for $D$. Moreover, the fact that the union of the rectangles has full measure follows from regularity of $\pi$?
- Assume $\mathfrak{m}$ to be locally finite. Then the author says that from the Lindelof property and separability there exists a Borel probability measure on $\mathfrak{h}$ such that $$\mathfrak{m}\ll\mathfrak{h}\leq C\mathfrak{m}\qquad\mathrm{for }\,C>0$$ and that $\frac{\mathrm{d}\mathfrak{h}}{\mathrm{d}\mathfrak{m}}$ is locally bounded from below by a positive constant. Do you have any hints on how to construct the measure $\mathfrak{h}$?
(the paper is about Optimal Transport in Lorentzian lenght spaces)