Let be a measure space. Let
be a measurable function. Suppose that
.
(a) Show that the set is of
-measure 0. (Intuitively, this is quite obvious, but we need to prove it rigorously.)
(b) Show that the set is
-finite with respect to
. i.e. it is a countable union of measurable sets of finite
-measure.
We may use Markov’s inequality, which turns out to be very useful in this question.
Proof: (a) Let , where
. Denote
.
, and
. (Markov Inequality!)
Then
Therefore, .
(b) Let ,
.
Therefore, , and we have expressed the set as a countable union of measurable sets of finite measure.
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