Well, you answered for CAT(0) but I also remarked that I would need this result for building.

@MarkSapir: Ok, you convinced me. But what about for Euclidean building, which is the context I’m actually interested in?

I wanted to use an argument like this, but I’m worried that this apartment will not contain the whole ray $\rho$. Assuming $x$ and $y$ are in the a common chamber, can I always find an apartment containing $y$, $x$ and $\rho$?

Thanks for your answer. What if I weaken the statement by only requiring the angle not to be acute, so allowing $\pi/2$?

In view of HJRW’s answer, is the statement true if I change obtuse with not acute (so I allow $\pi/2$)?

X is locally compact and Polish. Actually, all these measures in my problem are area measures of (singular) Riemannian metrics on X. I wasn't completely sure that my assumptions guarantee convergence of the areas.

The definition of tightness I found is the following: a sequence $\mu_{n}$ is tight if for every $\epsilon>0$ there is a compact set $K$ such that $\mu_{n}(K^c)<\epsilon$ for all $n$. The difference with what I have is that for me this condition holds only for $n \geq n_{0}$ with $n_{0}$ depending on $\epsilon$.