2018 Fall Real Analysis
Problem 1.
Let
Proof.
The set of divergence can be represented by
So it is measurable.
Problem 2.
Let
Proof.
We write
Then we have
We use Hölder Inequality to get
Thus we have
The theorem is proved.
Problem 3.
Let
(a).
(b).
Proof.
We use Eogorov's Theorem.
For any
and
This proves part (a).
For part
Thus we have
For the second term of the above right, we have the following estimate
For the first term of the above right, since
This completes the proof of part (b).
Problem 4.
Let
for all
Prove that there are
Proof.
For any continuous function
Define numbers
and
Then
On the other hand, we have
By definition of
The (3) is
Problem 5.
Let
converges absolutely for almost all
Proof.
For any
where
Thus
As a result, for almost all
Problem 6.
Let
where
Proof.
If
If
Therefore we have
Since
we have
Thus for almost all