propositional Modal logic filtration definition Hello I have a slightly unusual question which relates to a definition of filtration structure. The following is my current state of the definition:

$ \mathcal{M} = (W, R, L) $, W is a set of worlds,
$R$ is a binary relation on the worlds, $ R \subseteq W \times W$, and
$L$ is a labelling function $ L : W \rightarrow\ \mathcal{P} (PropAtoms) $, here $PropAtoms$ is the set of atomic formulas.

Let $ \mathcal{M} = (W, R, L) $ be a Kripke model and let $\Gamma$ be a set of formulas that is closed under subformulas. Now, for each world $ w \in W$ we define the following: 
$ \Gamma_{w} = \{ \psi \in \Gamma : (\mathcal{M},w) \vDash \psi \}$,

that is, $\Gamma_{w} $ contains formulas from $ \Gamma $ that are true at $w$ and consequently in model $\mathcal{M}$. Next we define an equivalence relation $\sim_{\Gamma} $ on worlds $w, w' \in W$, 
$ \sim_{\Gamma} w w' $ if and only if $\Gamma_{w} \sqsubset \Gamma_{w'}$,
that is, worlds $w$ and $w'$ are equivalent if and only if they satisfy the same formulas in $\Gamma$. We also have equivalence classes $[w]$ of worlds $w$ with respect to $\sim_{\Gamma}$.

Lets now present the following quotient structure $ \mathcal{M}_{\Gamma} = (W_{\Gamma}, R_{\Gamma}, L_{\Gamma})$ which is in fact filtration of $\mathcal{M}$ with respect to $\Gamma$. $\mathcal{M}_{\Gamma}$ is defined as follows:
.........

Here is the bit I need help with, I am struggling to define my filtration model. This is mainly because the definitions online don't use the same labelling function but rather use a valuation function which is of the form $v(p)$. And such function takes an atomic formula whearas my labelling function takes in a specific world written $L(x)$. Could someone please help me our here and try providing correct definition which will match my model $M$ definition?
You are probably looking for something like:
$L_\Gamma([w]) = L(w) \cap \Gamma$. This will falsify any propositional letter not in $\Gamma$.

(Filtration is (afaik) usually defined as a property that a model may have. There could be several (or none) concrete models satisfying this property.)