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The following proof is shown in my book too. Basically it states that every measureable function can be written as the pointwise limit of a sequence of simple functions. Now, my problem is I don't see where the measureability of the limit-function is used crucially - so is the measureability really strictly necessary for the proof and if so where is it used? For me the statement might as well be all functions can be written as the pointwise limit of a sequence of simple functions?