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Summary: 2 Lebesgue integration
1. Let (, A, µ) be a measure space. We will always assume that µ 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
,
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