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Summary: 4 Sequences of measurable functions
1. Let (, A, µ) be a measure space (complete, after a possible application of
the completion theorem). In this chapter we investigate relations between
various convergences of sequences of A -measurable functions {fn} on it.
2. Let us recall the various notions of convergence. Suppose f , f1, f2, . . .
are measurable.
(a) Convergence µ -almost everywhere (in particular the pointwise con-
vergence): fn f µ -a.e. in as n iff for some E with
µ(E) = 0 , we have fn(x) f(x) for any x \ E as n .
(b) Convergence uniform on E : fn f uniformly on E as
n iff
lim
n
sup
xE
|fn(x) - f(x)| = 0.
(c) Convergence in the Banach space Lp
, 1 p : fn f in Lp
as n iff f, fn Lp
and fn - f p 0 as n .
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