Let's denote by $JI(L)$ the set of join-irreducible elements of a lattice $L$, i.e elements $g\in L$ such that for any $a,b\in L$, if $g=a\vee b$ then $g\in \left\{a,b\right\}$. Let's say that $L$ is a $\alpha$-good lattice iff
1) It's equipped with a probability mesure, such that principle filters are mesurable sets, and every singleton has mesure $0$.
2) Any element of $L$ is the lower upper bound of some subset of $JI(R)$ that cardinality is strictly smaller than the continuum
Does there exist a good lattice $L$ such that for any $g\in JI(L)$, almost every element of $L$ is greater than $g$ ?
Note that if $JI(L)$ is a countable subset of $L$ the answer is no.
Motivation Let's say that a finite lattice $(L,\leq)$ is an $\epsilon$-lattice if there exists $g\in JI(L)$, such that the (uniform) probability that a random element of $L$ is grater than $g$ is lower then $1-\epsilon$. The Frankl conjecture says that any finite lattice is a $1/2$-lattice. Even the question : "does there exists $\epsilon>0$ such that any finite lattice is an $\epsilon$-lattice" is an open question for $\epsilon\leq 1/2$. (we will call it the WFC (weak Frankl conjecture). If WFC is wrong than we can find $(L_i,\leq ,0_i)_{i\in \mathbb N}$ and equip $L:=L_0\times L_1\times L_2....$, with the product mesure $\mathbb P$ of the uniform probabilty mesure on each $L_i$, in such a way that for any $g=(g_0,g_1....)\in JI(L_0)\times JI(L_1)...$, we have $sup_{j\in \mathbb N}\mathbb P(x\in g^j)=1$ (***)
where $g^j= \left\{x\in L,\, (0_0,0_1,...0_{j-1},g_j,g_{j+1}...)\leq x\right\}$. If we consider $M$ the sublattice of $L$ such that no projetion on some $L_i$ is $0_i$, and if we say that for any $a,b\in M$, $a\leq_M b$ iff $a^k\leq b$ for some $k\in \mathbb N$, then the quotient $M^*$ obtained from $M\cup\left\{0\right\}$ by identifying $a$ and $b$ iff $a\leq_M b$ and $b\leq_M a$, is a good lattice when you equip it with the $\mathbb P^*$ in such a way that $\mathbb P^*(A)=\mathbb P(\left\{x\in L,\,\exists y\in A,\,\, y\leq_M x\,\,and\,\, x\leq_M y\right\})$. And if I'm not mistaking, the (***) condition implies that for any $g\in JI(M^*)$ we have $\mathbb P^*(x>g)=1$.
[the problem is that $M^*$ is not generated by $JI(M^*)$ (it is not even complete as a lattice), so in order make the question fit better with the motivation, one should put instead of condition 2) a weaker hypothesis and just ask "new good lattice" to satisfy $\sup M^*=\sup K$ for some $K\subset JI(M^*)$, such that $|K|<|\mathbb R|$. I opened a new post, more specific to the link between the (same) motivation and a question (a bit different) : Probability , Martin Axiom, and the "weak" Frankl Conjecture