Let $(L,\leq,0,1)$ be a lattice, and let's denote by $JI(L)$ the set of its *join-irreducibles* (i.e. elements that are not the lowest grater bound of two **other** elements). We suppose that $\sup JI(L)=1$.
and that $\mathbb P$ is a probability on $L$ such that *ideals* and *filters* are measurable and singleton have probability 0. Let's call such a lattice a "reach-irreducible lattice" (RIB). If the cardinality of $L$ is that of the continuum" we can say "continuum reach-irreducible lattice" (CRIL)

Does there exist in any $L$ that is a continuum-reach-irreducible lattice, $g\in JI(L)$ such that $\mathbb P(\left\{x\in L,\, g\leq x\right\})<1$

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Related discussions on the MSE are https://math.stackexchange.com/questions/2737618/lattice-generated-by-join-irreducible-elements-such-that-any-principal-filter-c and https://math.stackexchange.com/questions/2742357/probability-martin-axiom-and-the-weak-frankl-conjecture

The first link is an imperfect attempt to fit with the *motivation* of this question, that is explained in the second link. If the answer to this question is yes, then a weak version of the Frankl Conjecture ("WFC")- that is also an open problem - is true (see MSE links).

Note that in both links, the lattices that I built assuming that WFC is wrong are not exactly CRIL, but up to small details, one can easily get CRIL from them.

Note also that the cardinality condition is just given to fit with WFC, but it would be nice indeed if the answer of the question is yes for any RIL. And if it is not, it might also give some useful informations, so I would also be happy to read answers and comments about general RIL.