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.