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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