# Modular which is metrizing but does not satisfy the $\Delta_2$ condition

Let $\Phi$ be a nice Young function (N-function) and $(\Omega,\mathcal{F},P)$ a probability space such that either $P$ is diffuse on a set of non-zero probability or $P$ is purely atomic and there are infinitely many atoms. In other words $(\Omega,\mathcal{F},P)$ is not a finite probability space.

It is said that function $\Phi$ satisfies the $\Delta_2$ condition at infinity if there is $K>0$ and $u_0\ge 0$ such that $$\Phi(2u)\le K\Phi(u) \text{ for all }u\ge u_0.$$

It is said that modular $m$ given by $m(X):=E[\Phi(X)]$ is metrizing if for every sequence of random variables $\{ X_n\}$ one has that $m(X_n)\to 0$ implies $m(2X_n)\to 0$.

Give an example of a Young function that does not satisfy the $\Delta_2$ condition, together with a (purely atomic but infinite) probability space on which the corresponding modular is metrizing; or, prove that such an example does not exist.

It is known that

1. such an example does not exist if $P$ is diffuse on a set of non-zero probability;
2. such an example does exist on a purely atomic space with infinitely many atoms if the measure $P$ is allowed to be infinite (not a probability measure anymore);
3. on a finite probability space every N-function modular is trivially metrizing regardless of the $\Delta_2$ condition on $\Phi$.