Interpretation of sufficient condition for tightness on $C[0,1]$ I try to find the right way to think about the following equality, for all $\epsilon$,
$$\lim_{\delta \rightarrow 0}\limsup_{n} P_{n}(x; w(x,\delta)\ge \epsilon)=0 $$
where $P_{n}$ is a sequence of measures on $C[0,1]$ and $w(x,\delta)=sup_{\mid s-t \mid \le\delta}\mid x(s)-x(t)\mid$.
It is also expressed as for all $\epsilon$ and for all $\eta$ there is a $\delta$ and $n_0$ such that
$$P_{n}(x; w(x,\delta)\ge \epsilon)< \eta$$ for all $n \ge n_{0}$.
My own idea so far is the following,
Far enough into the sequnce, the measure of the set where uniform equicontinuity falis has measure zero.
1.A thing that stuck me was that it donst seem that we can commute quantifiers, even tho they come in the order $\forall \epsilon ,\eta \exists \delta ,n_{o}$.
$$\limsup_{n} \lim_{\delta \rightarrow 0}P_{n}(x; w(x,\delta)\ge \epsilon)=0, $$
should hold for all measures on the space $C[0,1]$. 
I.e it donst seem that this is expressed properly the second way.
2.Why dont we just put $> 0$ instead of for all $\epsilon$ inside the measure? Answered.
Im reading Billingsleys book 2 ed, Convergence of probability measures page 82.

 A: Edit : If you take $\limsup_{n\to \infty} \lim_{\delta \to 0} \Bbb{P}_n (\omega_x(\delta) > \epsilon) = 0$, the part $\lim_{\delta \to 0} \Bbb{P}_n (\omega_x(\delta) > \epsilon) = 0$ would always be 0 since under $\Bbb{P}_n$ each x is in $C$. And you would have to evaluate a limsup of zeros which is zero itself. So this order of limiting really characterize any sequence in $C$.
On the other hand if you take  $ \lim_{\delta \to 0} \limsup_{n\to \infty} \Bbb{P}_n (\omega_x(\delta) > \epsilon) = 0$ that means you are controlling the probability of straying away of equicontinuity over all your adherence values (combined with condition one actually). (I hope it is less confusing...) 
To answer your first question, if you swap the limits, it would mean equicontinuity no more and it would be always true for any sequence in $C$ . And what we want is to caracterize equicontinuous sequences not just any one valued in $C$. 
A great way to see this is to look at the definition of equicontinuity. Equicontinuity means you have some kinda uniform control over the variation of all your functions alltogether. While swapping the limits gives you control over each function independantly of the others which makes them just a sequence of continuous functions.
(Sorry im on mobile so it's kinda hard to use mathematical signs) 
