The question how to show convergence in probability imply convergence a.s. in this case? uses a result called Ottaviani's inequality. Where can I learn about the original Ottaviani's inequality, and the Levy-Ottaviani's inequality?
I have searched it so long but have no useful results. You may give some links or some books about it. Thank you.
I Think here: https://link.springer.com/content/pdf/bbm%3A978-1-4757-2545-2%2F1.pdf would be the best.
In A.1.1 and A.1.2, it stated and proved the Ottaviani's Theorem then Levy's inequality consecutively.
The second one is commonly named as Ottaviani-Levy's inequality.
I believe there are some other invariances of these two theorems but the proof technique would be the same. You could see the proof of Levy is really similar to the proof of Ottaviani.
Oh by the way, it has different notation, so by $\mathbb{P}^{*}$, I think we can just treat it as the regular probability measure $\mathbb{P}$, and by $\|\ \ \|^{*}$, I believe it is just absolute value $|\ \ |$, and by $\|\ \|$ the book clarifies it as $\sup|\ \ |$.
I believe the $*$ using here must deal with some outer measure and outer a.s. convergence, but in the case of your link, the outer measure is clearly the measure itself, and the outer a.s. convergence is therefore the regular a.s. convergence.
