So, my book says that since measurable functions are closed under monotone limits, this means that the collection is closed under all pointwise limits. It goes on to say that this is the evidence that the class of measurable functions is quite large.
Can somebody help me to understand why the collection of measurable functions being closed under point-wise limits is evidence that the collection of measurable functions is large?
Furthermore, the book leaves it as an exercise to prove that sequences of measurable functions are closed under a.e. point-wise convergence, but I can not figure out the proof.
I'd really appreciate some help!!