# “A Master Dominated Convergence Theorem” From Gordan Žitkovic's Probability Theory notes

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 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.

• 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. – Clement C. Aug 22 '17 at 0:15
• 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? – User086688 Aug 22 '17 at 0:31
• See e.g. this (Theorem 4.5, no proof as far as I can see) or this (p.4). – Clement C. Aug 22 '17 at 0:33
• 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. – User086688 Aug 22 '17 at 0:39
• Sure. You can also have a look at Theorem D of Paul R. Halmos, Measure Theory, D. Van Nostrand, New York, 1950. – Clement C. Aug 22 '17 at 0:46