We know that probability measures are tight if the metric space is separable and complete.
Here tight means there exists a compact set in that metric space say $K$ such that $P(K) > 1- \epsilon$.
I want to create a probability measures which is not tight. For that we have to violates the separability or completeness condition. Suppose we violates separability. And consider the space $l_{\infty}$ with respect to supremum norm. We know that this space is not separable. But how to construct an probability measures here?
If we violates completeness then also this holds. As a example $c_{00}$ space is not complete. Then how to construct probability measures there?
Any kind of simple examples are appreciated.
I know there is some explanation and example available in the stack exchange and those are talking about 'left limit topology' kind of things. I need an simple example and construction not that much advance that's why asked this question.
$c_{00}$ is an increasing union of subsets $K_n$ where $K_n$ consists of sequences where only the first $n$ elements are allowed to be nonzero. Thus for every probability measure on $c_{00}$ and every $\epsilon>0$, there is some $n$ such that $P(K_n)>1-\epsilon$. So that space will not yield an example.
Every Borel probability measure on a metric space has a separable support, see Lemma 2.1 in https://arxiv.org/pdf/1612.03213.pdf But whether this support has full measure is more delicate.