The function spaces Lp(μ) , L(μ) , Cc(X), and C0(X)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 19 March 2011

The Banach spaces Lp(μ), L(μ), and C0(X)

Let (X,) be a measurable space and let μ: [0,] be a positive measure on .
Let p>0 and let f:X be a measurable function. The Lp-norm of f is

fp = (X |f| p dμ ) 1/p .
Define
Lp(μ) = { f:X | fis measurable and f p < }

Let f:X be a measurable function. The essential supremum of f is

essupf = inf { α | μ( f-1 ((α,]) ) =0}
with essupf =  if { α | μ( |f|-1 ((α,]) =0} = .

Let f:X be a measurable function. The L-norm of f is

f =essup|f| .
The essentially bounded measurable functions are the elements of
L(μ) = { f:X | fis measurable and f < }

Let (X,) be a measurable space and let μ:[0,] be a positive measure on .

(a)   If p1 then Lp(μ) is a complete metric space with respect to p.
(b)   L(μ) is a complete metric space with respect to .

Let X be a locally compact Hausdorff topological space. Let f:X be a continuous function. The support of f is

supp(f) = {xX | f(x)0} ,
the closure of {xX | f(x)0} . Define
Cc(X) = { f:X | fis continuous and supp(f) is compact}
A function f:X vanishes at infinity if f satisfies
if ε>0 then there exists a compact set KX such that if xK then |f(x)|<ε.
Define
C0(X) = { f:X | fis continuous and f vanishes at infinity}
Define : C0(X) >0 by
f = sup{ |f(x)| | xX} .

Let X be a locally compact Hausdorff topological space.

(c)   C0(X) is a complete metric space with respect to with .

HW: Give an example showing that Cc(X) is not always a complete metric space with respect to with .

HW: Show that if X=k and μ is Lebesgue measure then = giving that L(μ) = C0( k).

Let X be a locally compact Hausdorff topological space. Let μ be a regular positive Borel measure on X.

(a)   The space C0(X) is the completion of Cc(X) with respect to .
(b)   If p1 then the space Lp(μ) is the completion of Cc(X) with respect to p.

HW: Give an example showing that The space L(μ) is not necessarily the completion of Cc(X) with respect to .

Notes and References

These notes were written for a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, Chapters 3]. See [Ru, Chapt. 3 Ex 21] for the resolution of the issue of completion vs dense subsets.

References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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