Folland claims that the space of a.e.-defined integrable functions form a complex vector space (under pointwise a.e. addition and scalar multiplication). How do you show this? Also, this definition seems very awkward to me, since you cannot conveniently take limits or infinite sums of measurable a.e.-defined functions and stay assured that the result is another measurable a.e.-defined function. I have trouble understanding this whole idea about a.e.-defined integrable functions and operations on them. Thank you in advance for your kind help!
2026-03-25 17:17:04.1774459024
a.e-defined integrable functions
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To show that it is a vector space you just need to make sure that the sum of two elements and the scalar multiple of an element belong to it. By induction, it follows that all finite sums and, in general, all finite linear combinations belong to the space. You do not have to worry about limits or infinite sums. Those only make sense if you have an additional topological (metric, etc.) structure.