Prove that is not a limit of increasing step functions.
Let be sequence of connected bounded subsets (interval) of real numbers. Step function is defined to be the finite linear combination of their charateristic functions.
while for .
Let be a sequence of open intervals in (0,1) which covers all the rational points in (0,1) and such that . Let and show that there is no increasng sequence of step function such that almost everywhere. (by means of increasing, for all x)
I think is what author intended. is decreasing and . this shows that converges. so almost everywhere. However convergence of doesn't prove the non-existence of increasing step function which converges to f a.e.
How can I finish the proof?