metric on Positive Borel measures and absolutely continuity w.r.t to Lebesgue measure. Consider the following distance between nonnegative borel measures on $[0,T]$
$\qquad \qquad \qquad d(\eta,\nu) := \sup_{s\leq T}|\eta([0,s]) - \nu([0,s])|$ 
Let $\eta^n$ a sequence of lebesgue-absolutely continuous measures with density $h^n$ such that $\int_0^T h^n(s)ds < K \quad \forall \: n$, 
First question  if $\eta$ is such that  $\quad \lim\limits_{n} d_T(\eta^n,\eta) = 0 \quad $, then $\eta$ is absolutely continuous w.r.t Lebesgue measure ? 
A question i had in mind: Does this metric is well known in the literature? ( i wonder if it metricizes the weak convergence on space of positive measures such as Wasserstein one ) 
Sorry if i'm a little bit evasive. I'm not used to this topic so i really need help or references to read. 
 A: This is, when specialized to probability measures, the Kolmogorov Smirnov distance.  It does not metrize weak* convergence: let $\mu_n$ be a point mass at $1/n$ and let $\mu$ be a point mass at $0$.  Then $\mu_n\to\mu$ weak* but $d(\mu_n,\mu)=1$ for all $n$.  
Intuitively, the metrics that metrize weak* convergence of probability measures, such as the Wasserstein and Levy-Prokhorov metrics, measure a combination of the sideways and vertical distances between the distribution functions whereas the KS metric only measures vertical distance. (If I were at a blackboard I'd wave my hands to make clear what sideways and vertical mean.)  
I think one can write a sequence of measures with densities converging to Cantor measure, which would make the answer to your first question no.  Here is a sketch.  Consider the usual representation of a Cantor distributed random variable $X=\sum_{k>0} B_k / 3^k$, with iid Bernouilli bits $B_n$ for which $P(B_n=0) = P(B_n=1) = 1/2.$  Write $X_n = \sum_{k=1}^n B_k / 3^k + U/3^{n+1}$ where $U$ is uniformly distributed. Equivalently, let $$X_n=\sum_{k=1}^n B_k 3^{-k} + 3^{-(n+1)} \sum_{k>0} B_{n+k}  2^{-k}.$$ Let $\eta$ and $\eta^n$ be the distributions of $X$ and $X_n$, and so on.
