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?