I’m reading “Optimal Transport Theory for Applied Mathematicians” by Santambrogio, and in the proof of one of the Theorems, the author states the following:
“Fix $\epsilon>0$ and find two compact sets $K_x \subset X$ and $K_y \subset Y$ such that $\mu(K_x^c), \nu(K_y^c)< \epsilon/2$ (this is possible thanks to the converse implication in the Prokhorov theorem, since a single measure is always tight).”
My question is if the statement in the parenthesis is indeed correct. I mean, if we are talking about probability measures, or any limited measure, then I can clearly see how they are indeed tight. But it seems to me that not every single measure is always tight. Can anyone show how indeed every single measure is always tight? Assume that $X$ and $Y$ are Polish.