3
votes
1answer
119 views

Please check if my proof is correct of Monotone Convergent theorem

I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT. The hint I was given is let $f_n$ be a sequence of increasing function, ...
0
votes
1answer
53 views

Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
1
vote
3answers
215 views

Fatou's Lemma Strengthened to Equality

I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional ...