#Measurability of sup of measurable functions

30 messages · Page 1 of 1 (latest)

potent summit
#

I don't understand this step in a proof in Adult Rudin, Theorem 1.14.

Let $X$ be a topological space and let $f_n: X \to [-\infty,\infty], n = 1,2,3,\ldots$ be a sequence of measurable functions.
Rudin proves that $g = \underset{n \ge 1}{\sup} \ f_n$ is measurable.
The proof uses:
$$g^{-1}((\alpha,\infty]) = \bigcup_{n=1}^{\infty} {f_n}^{-1} ((\alpha,\infty])$$ for all $\alpha \in \mathbb R$, why does this equality hold?

uncut ridgeBOT
#
  1. Do not ping the Moderators, unless someone is breaking the rules.
  2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
  3. Do not ping other members randomly for help.
  4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  5. Wait patiently for a helper to come along.
  6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

pseudo lichenBOT
#

gfauxpas

205675981845954560.png
visual spire
pseudo lichenBOT
#

Omegabet_

186109587366215691.png
visual spire
#

$g^{-1}((\alpha,\infty])={x\in X|g(x)>\alpha}={x|\sup_n{f_n(x)}>\alpha}$

pseudo lichenBOT
#

Omegabet_

186109587366215691.png
visual spire
potent summit
#

ah, thank you!

#

The top voted answer seems to have a mistake of confusing the definition of lim sup with sup, but I should be able to derive the correct answer from it

#

oh wait no, I'm wrong, the answer is correct

potent summit
#

@visual spire how do I award you points?

visual spire
#

I think just say thanks?

potent summit
#

thanks!

ivory rapids
#

you can thank them, and then click the red button to close the post

#

The rep bot is permanently down to prevent abuse, so it does not react to the keyword "thanks" anymore

potent summit
#

@ivory rapids didn't work

#

no red button or prompt

ivory rapids
#

Ok now that is just weird

#

the +close command does not work?

#

It seemed to work for me

#

@potent summit

potent summit
#

oh I was using the hotdog button close

#

+close

gleaming gulchBOT
# potent summit +close
Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

gleaming gulchBOT
# gleaming gulch

Thank you for your feedback! Omegabet_ has been awarded 1 helper_points. They now have 822 helper_points. They have 3 helper_points daily left for today.

ivory rapids
#

now the red button will close the post