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I am trying to review the way L^p spaces are treated differently

in Royden. In Ch.6, he treats them under "Classical Banach Spaces",

and then again, in his Ch.11 , under "Abstract Spaces".

This is what I understand: (Please comment/correct)

In the case of abstract spaces, we do not deal with R^n,

or subsets of R^n, where the idea of measure is "natural" , and

where we start by defining the measure of an open "box" ( a

subset bounded by open intervals (a1,b1),(a2,b2),..,(an,bn))

which we define to have measure (a1-b1)(a2-b2)...(an-bn)

and build our measure around this, by defining the measure of a set

using the infimum of coverings by these boxes.

In abstract spaces, instead of R^n, we choose any topological space,

then define a sigma algebra , like, e.g., the Borel algebra generated by

open sets, or one generated by the compact subsets, etc. , which we

define to be the collection of measurable sets.

Then we define measurability in terms of being a member of

the sigma algebra, and , given abstract spaces X= (X,s_X,mu_X), Y=(Y,s_Y,mu_Y) , with

s_X,s_Y sigma algebras and mu_X,mu_Y measures (satisfying the basic

axioms ;subadditivity, monotonicity eyc.) , we then define f:X-->Y ; X,Y as above

is measurable, if for U_y in s_Y (the sigma -algebra in Y ), f^-1(U_Y) is in s_X (the

sigma-algebra of X).

What I am not clear on, is the issue of integrability: in abstract

spaces, we integrate against a measure. How is this different from

Lebesgue integration, where we do a sum :Sumf(y)( mu_X (x):f(x)=y)

i.e., multiply any value f(y) in the range by the measure of its preimage?.

It seems to be the same?.

Question:

Is any of these abstract spaces , or measure triples (X, s_X,mu_X) a Hilbert space,

or are they all Banach spaces?. Are there also non-measurable subsets in these

abstract spaces?.

Thanks for Comments/Corrections/Examples/Refs

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# Measure theory in R^n and in abstract spaces

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