# Convergence in measure implies convergence in $L^p$ under the hypothesis of domination

Given a sequence $f_n \in L^p$ and $g \in L^p$, with $|f_n| \leq g$, I am trying to show that $f_n \to f$ in measure implies $f_n \to f$ in $L^p$.

Firstly, I know that if $f_n \to f$ in measure, then there is a subsequence $f_{n_i}$ such that $f_{n_i} \to f$ almost everywhere. Then I can use the dominated convergence theorem to show that $\lVert f_{n_i} - f_p\rVert \to 0$.

Now I am trying to show that $\lVert f_n - f\rVert_p \to 0$. My idea is to assume that $\lVert f_n - f\rVert_p \nrightarrow 0$ and then show that this contradicts the fact that $\lVert f_{n_i} - f\rVert_p \to 0$, but I am not sure of the details. Can anyone help me finish the argument?

• In the title, you should add "with an hypothesis of domination". You are almost done: if $||f_n-f||_p$ doesn't converge to $0$, then we can find $\delta>0$ and a subsequence such that $||f_{n_k}-f||\geq \delta$. This subsequence still converges in measure to $f$, so by your previous argument we get a contradiction. Oct 18, 2012 at 19:01
• @DavideGiraudo Ok, got it. Thanks!
– rt93
Oct 18, 2012 at 19:16
• You can answer your own question (hence it won't remain unanswered), and you will have your homework done properly. Oct 18, 2012 at 19:17
• @DavideGiraudo : as one commenter makes note of below, we haven't used the boundedness of $f_n$ at all. Is this a problem?
– Alex
Aug 8, 2017 at 18:08
• Also, this question was featured on the UI-Urbana Champaign graduate qualifying exam from August 2015, if anyone cares to know.
– Alex
Aug 8, 2017 at 18:09

If $\lVert f_n - f\rVert_p \nrightarrow 0$, there exist a subsequence $f_{n_i}$ such that $\| f_{n_i} - f\| _p \ge \varepsilon$ for some $\varepsilon >0$. But $f_{n_i}$ still converges in measure to $f$. So, again There is a subsequence $f_{n_{i_j}}$ of the $f_{n_i}$ such that $\|f_{n_{i_j}} - f\|_p \rightarrow 0$, and this is a contradiction.
• How does the boundedness of $\|fn\|$ work in your argument? Aug 10, 2016 at 3:30
• To claim the fact that $\|f_{n_{i_j}} - f\|_p \rightarrow 0$ you would need to apply Dominated convergence theorem in $L^p$ setting, there you would need boundedness of $f_n$ Feb 11, 2017 at 6:56