The Lebesgue Convergence Theorems

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

Last updates: 2 April 2011

The Lebesgue Convergence Theorems

(Lebesgue's monotone convergence theorem)
Let (X,) be a measurable space and let μ: [0,] be a positive measure on (X,).
Let fn:X [0,], n>0, be a sequence of measurable functions such that

(a)   If xX then 0 f1(x) f2(x) , and
(b)   If xX then limn fn(x) exists.
Let f:X [0,] be given by f(x) = limn fn(x) .
Then f is measurable and
X ( limn fn ) dμ = limn ( X fn dμ ) .

(Fatou's lemma)
Let (X,) be a measurable space and let μ: [0,] be a positive measure on (X,).
Let fn:X [0,], n>0, be a sequence of measurable functions. Then

X ( liminfn fn ) dμ liminfn ( X fn dμ ) .

(Lebesgue's dominated convergence theorem)
Let (X,) be a measurable space and let μ: [0,] be a positive measure on (X,).
Let fn:X , n>0, be a sequence of measurable functions such that

if xX then limn fn(x) exists.
Assume that there exists g L1(μ) such that
if n>0 and xX then |fn(x)| g(x) .
Then
(a)   fL1(μ),
(b)   limn ( X |fn-f| dμ ) =0 ,
(c)   X ( limn fn ) dμ = limn ( X fn dμ ) .

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 1-6].

References

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

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