Is it really necessary to consider measurable banach space mappings?
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What is needed is only scalar measurability that is measurability of | f| for forming l1 space and L2 space.
Since issues of measurability depend upon axiom of choice it is advisable to avoid these issues as far as possible.
It is well known that in any integration theory ( daniell, henstock kurzweil lebesgue bochner) each absolutely integral real valued function is measurable.
But in daniell mikisuinski or henstock kurzweil integral one does not need prior measurability for discussing integral
l1 can be defined as space of absolutely integrable mappings L2 as space of mappings such that | f| square is integrable.
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