2
$\begingroup$

I am using the H.L Royden 4th ed and the Folland

Any definition of measurable functions or otherwise is as defined in the text above

1.Let {$f_{n}$} be a sequence of measurable functions converging a.e(almost everywhere) to the function $f$ on $E$. Prove that $f$ is measurable.

2.Let ($\Omega$, $\Sigma$) be a measurable space. If $f$ is a pointwise limit of a sequence {$f_{n}$} of measurable functions on a common domain $D$ $\in$ $\Sigma$. Then $f$ is measurable.

My attempt For 1. I choose a set say $A_{0}$ such that this set is a subset of the entire set say $A$ . Next i proved that the measure of the first set is 0 and showed that $f_{n}$ converges pointwise to the compliment of the first set.

Thereafter we know that f is measurable if and only if the restriction of the set is measurable. I then say that from this I assumed that the convergence to all possible sets A is pointwise.

Then choose an element say b inside the entire real line and used the definition of pointwise convergence so show that f(x) is less than the element say b. And after some working with these sets I showed that the intersection of the entire set belongs to $M$. After which the union of my entire set such that b is inside A is equal to my f(x) being less than b which shows that f is measurable since the countable union of measurable sets is again measurable so my LHS is measurable and my RHS is measurable so hence f is measurable.

For 2. I have no idea how to even begin attempting the question. What my thought process?I have never done a question with ($\Omega$, $\Sigma$). Can anyone point me in the right direction how to begin such a question.

$\endgroup$
  • $\begingroup$ what is M in the second to last paragraph? $\endgroup$ – user24142 Oct 26 '18 at 4:57
  • $\begingroup$ If the measure space is not complete (in the sense of measure theory), then the first statement is false, because you can modify $f$ outside the set of convergence such that $f$ is not measurable. (Just take a Vitali-like set!) $\endgroup$ – p4sch Oct 26 '18 at 6:38
0
$\begingroup$

For (1), I would suggest the enhanced version of the same book by Royden-Fitpatrick the fourth edition. You will find a full explanation of each step which will help you even to understand your own proof.

For (2), I think you can benefit from the following link which asks the same question but with a minor difference: Measurability of a pointwise limit of measurable functions

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.