Let $\mathcal{P}$ be the set of measures on $(\mathbb{R},B(\mathbb{R}))$, for $\mu, \sigma \in \mathcal{P},$ let $d(\mu,\sigma)=\inf(\epsilon>0;\forall x \in \mathbb{R},F_{\sigma}(x-\epsilon)-\epsilon \leq F_{\mu}(x) \leq F_{\sigma}(x+\epsilon)+\epsilon)$ where $F$ denotes the distribution function of a measure.
We can prove that $(\mathcal{P},d)$ is a metric space and that $(\mu_n)_n$ converges weakly to $\mu$ if and only if $\lim_nd(\mu_n,\mu)=0$.
I am studying some properties of this space such as completeness and separability.
For the separability, if we take $\mu \in \mathcal{P},$ and $\mu_n=\sum_{k \in \mathbb{Z}}\mu([(k+1)/2^n;k/2^n[) \delta_{k/2^n},$ we have $(\mu_n)_n)$ converges weakly to $\mu,$ since for all bounded and continuous functions $f$ we have $$\int_{\mathbb{R}}f(x)d\mu_n(x)=\int_{\mathbb{R}}f(\left \lfloor{x2^n}\right \rfloor/2^n)d\mu(x)$$ Can we use this to prove that $(\mathcal{P},d)$ is separable?
For the completeness, if we take a sequence $(\sigma_n)_n$ such that
$\forall \epsilon>0,\exists n_0 \in \mathbb{N};\forall p \geq n_0,q \geq n_0,d(\sigma_p,\sigma_q) \leq \epsilon,$ and here I am totally stuck.