MATH 507/420 - Assignment #3
Due on Friday October 21, 2011
Name —————————————–
Student number —————————
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Problem 1:
Show that an extended real valued function f on X is measurable iff
for each rational number c, f −1 ([−∞, c]) is a measurable set.
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Problem 2:
Let fn be a sequence of real valued measurable functions on X such
that for each natural number n,
µ({x ∈ X s.t. |fn (x) − fn+1 (x)| > 1/2n }) < 1/2n .
Show that fn converges pointwise almost everywhere on X.
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Problem 3:
A sequence {fn } of real valued measurable functions on X is said to
converge in measure to a measurable function f provided that for each
r > 0,
lim µ({x ∈ X s.t. |fn (x) − f (x)| > r}) = 0.
n→∞
A sequence {fn } of real valued measurable functions on X is said to
be Cauchy in measure provided that for each r > 0 and > 0 there is
N ∈ N such that for all m, n > N ,
µ({x ∈ X s.t. |fn (x) − fm (x)| > r}) < .
(a) Show that if µ(X) < ∞ and {fn } converges pointwise a.e. on X to
a measurable function f then {fn } converges to f in measure.
(b) Show that if {fn } converges to f in measure, then there is a subsequence of {fn } that converges pointwise a.e. on X to f .
(c) Show that if {fn } is Cauchy in measure then there is a measurable
function f to which {fn } converges in measure.
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Problem 4:
Assume µ(X) < ∞. Show that fn → f in measure iff each subsequence
of {fn } has a further subsequence that converges pointwise a.e. on X
to f . Use this to show that for two sequences that converge in measure,
the product sequence also converges in measure to the product of the
limits.