functional analysis – Does \$L_1\$ Convergence imply almost everywhere convergence for the Set of all increasing functions on \$[0,1]\$ to \$[0,1]\$?

We know that $$L_1$$ convergence does not imply convergence a.e. in general. However, consider the following set
$${f:(0,1) rightarrow (0,1)mid f(x)geq f(y) quad forall xgeq y}.$$ Take any sequence $${f_n}_1^infty$$ from the set that is converging to $$f$$ with respect to $$L_1$$ norm. Is it possible that convergence in $$L_1$$ implies pointwise a.e. convergence in this case? I just couldn’t find an example to show this is not true.