Im looking at "A Master Dominated Convergence Theorem" Proposition 11.11 from Gordan Žitkovic's Probability Theory notes (its on page 145) and the link to his notes can be found here: https://www.ma.utexas.edu/users/gordanz/notes/theory_of_probability_I.pdf

I am stuck on the last part of his proof where he says that enter image description here

I know that in order to apply the bounded convergence theorem, we need $\Psi_M\left(\left\vert X_n\right\vert^p\right) \overset{a.s.}\longrightarrow \Psi_M(\left\vert X\right\vert^p)$. But we only have that $\Psi_M(X_n)\overset{\mathbb{P}}\longrightarrow \Psi_M(X)$ by the continuous mapping theorem...and I am not too sure how to get to the almost sure convergence given we only have the convergence in probability! Thanks for the help in advanced.

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    $\begingroup$ Well, as it turns out the bounded convergence theorem on ly requires convergence in probability. This is, sadly, not mentioned in the notes as far as I can see. $\endgroup$ – Clement C. Aug 22 '17 at 0:15
  • $\begingroup$ Oh really?!?! Is there any chance you could point me to a resource which proves the bounded convergence theorem but with only convergence in probability? $\endgroup$ – User086688 Aug 22 '17 at 0:31
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    $\begingroup$ See e.g. this (Theorem 4.5, no proof as far as I can see) or this (p.4). $\endgroup$ – Clement C. Aug 22 '17 at 0:33
  • $\begingroup$ Thanks for those sending me those links so quickly. There seems to be a very short proof in the second resource from Wharton. If I get stuck trying to understand this proof/trying to prove it myself, would you mind giving me a hint or two along the way? (I suppose I can happily set up a new question) for this. It just seems as though most of the resources that I google on the bounded convergence theorem only prove for the case of convergence almost surely or pointwise convergence. $\endgroup$ – User086688 Aug 22 '17 at 0:39
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    $\begingroup$ Sure. You can also have a look at Theorem D of Paul R. Halmos, Measure Theory, D. Van Nostrand, New York, 1950. $\endgroup$ – Clement C. Aug 22 '17 at 0:46

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