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2 Lebesgue integration 1. Let (, A, ) be a measure space. We will always assume that is com-
 

Summary: 2 Lebesgue integration
1. Let (, A, ) be a measure space. We will always assume that is com-
plete, otherwise we first take its completion. The example to have in mind
is the Lebesgue measure on Rn
, (Rn
, Ln, | |) . We will build the inte-
gration theory for A -measurable functions. We will consider measurable
functions
f : - R,
where R = R1
{-} {+} (and also functions F : C {} ,
where C = C{} ). First we define integrals of real valued nonnegative
functions, and then reduce the general case to this. We will follow Rudin
very closely.
2. Let f : R be a simple function:
f =
N
j=1
cjEj
,

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics