Talk:Measure Theory/Riesz' representation theorem

Unclear for me in step 2.

The proof says: «By definition $$ \Lambda g\geq \mu (K)$$». For me, that is not clear. The definition of $$\mu(K)$$ is the inf of $$\mu(V)$$ where $$V$$ is open with $$K\subset V$$ while, for each of these open sets, the definition of $$\mu(V)$$ is a supremum of $$\Lambda f$$.

Thus for each $$\alpha$$ I can say $$\mu(V_{\alpha})\geq \Lambda g$$ and $$\mu(K)\leq \mu(V_{\alpha})$$. But this is not sufficient.

Is there a continuity hypothesis on $$\Lambda$$ ?

> I found an answer here: https://4dspace.mtts.org.in/expository-article-download.php?ai=160 > It may be useful to add that as a link in the proof.


 * Yes please do that. -- Jules (Mrjulesd) 19:54, 5 April 2022 (UTC)


 * Well... I re-read the proposed proof, and it seems to me that some steps are missing even in the source I mentioned. I wrote a proof here : https://laurent.claessens-donadello.eu/pdf/giulietta.pdf (search for "THOooTWZWooHqGDAx" in the section about Haar measure). I don't know if I can add that document in the sources because 1. that's autopromotion 2. I have no secondary sources for the gap I fill.
 * I don't think that is likely to be a problem as long as you are sure it is correct. -- Jules (Mrjulesd) 10:25, 11 April 2022 (UTC)