Can you provide me a counter example for this.


Suppose that I have a sequence of probability measures
$(\mu\_{r,t})\_{r,t>0}$ on a compact space metric $X.$


Suppose additionally that:


there exists a probability measure 
$\mu\_{r,0}$ such that $\mu\_{r,t}\to^{\*,t\to 0}\mu\_{r,0}$ 


and


there exists a probability measure $\mu\_{0,0}$ such that $\mu\_{r,0}\to^{\*,r\to 0}\mu\_{0,0}.$


Is it true that there exist the weak\* limit of 
$\mu\_{r,t}$ as $r\to 0$ for every $t$ close to $0?$



A very natural question in weak* topology
Let $X=\{1,2\}$ and define $\mu\_{r,t}:= \delta\_1$ either if $r \geq t$, or if both $r<t$ and $\lfloor \frac{1}{r}\rfloor$ is even. Define $\mu\_{r,t}:=\delta\_2$ if both $r<t$ and $\lfloor \frac{1}{r}\rfloor$ is odd. Clearly $\lim\_{t \to 0}=\delta\_1$ for every $r>0$, and $\lim\_{r \to 0} \lim\_{t \to 0}\mu\_{r,t}=\delta\_1$ as hypothesised. It is equally clear that $\lim\_{r \to 0} \mu\_{r,t}$ does not exist for any $t>0$.


There is a heuristic underlying this answer: the set of all probability measures on $X$ in the weak-\* topology contains a homeomorphic copy of $X$ as a closed subset, embedded by the map $x \mapsto \delta\_x$. Thus if one can find a sequence $(x\_{r,t})$ of elements of $X$ with analogous convergence properties then the measures $(\delta\_{x\_{r,t}})$ will be a counterexample.


