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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!!

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    $\begingroup$ Please only ask one question per thread. Your first question is too vague to be answered as it stands. $\endgroup$ – tomasz Jun 19 '18 at 23:01
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The second statement is false, in general. You require a condition called completeness of the sigma algebra to say that a.e. limits of measurable functions are measurable. For the part recall that pointwise limits of continuous functions inlcude all continuous functions and some non-continuous functions also. If you take this new class and take pointwise limits you get an even bigger collections of functions, and so on. This should give you some idea as to why you get a large class if you start with, say, simple measurable functions and keep taking pointwise limits.

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