Suppose I have a parametrized sequence of functions $f(t)$ for $t \in [0,1]$ where each $f(t) \in L^2(\Omega)$ on some bounded domain $\Omega$.
If $f(t)$ is increasing in $t$ and uniformly bounded in $L^2$, does this imply that $f(t) \to f$ as $t \to 1$ for some function $f$?
I.e. does the monotone limit exist like it would do for countably indexed sequences?
If you are talking about pointwise convergence let $f=\lim f( 1-\frac 1 n)$ then $f(t) \to f$ as $ t \to 1$ by monotnicity and squeeze theorem.