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.

  • $\begingroup$ 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

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


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