# Prove convergence in $L^1$

Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Let $f_n \in L^1$. Assume $f_n \rightarrow f$ a.e. and there exist $p > 1$ and $c > 0$ such that $$||f_n||_p < c$$ for all $n$.

I want to show that $$f_n \rightarrow f \ \mbox{in} \ L^1, \ \mbox{i.e.,}\ \int_X |f_n - f| d\mu \rightarrow 0 \ \mbox{as} \ n \rightarrow \infty .$$

Since $f_n \rightarrow f$ a.e., I want to interchange limit and integration to get $$\lim \int |f_n - f| = \int \lim |f_n - f| = 0.$$ I think that the bounded condition $||f_n||_p < c$ might be used to obtain an integrable bound for $g_n = |f_n - f|.$ Then I will apply Dominated Convergence to interchange integration and limit. However, it seems that this approach does not work.

Any suggestion to solve the problem ?

• You can show $(f_n)$ is uniformly integrable and appeal to Vitali's Theorem. – David Mitra Dec 1 '15 at 11:55
• @DavidMitra So, since $$||f_n||_p < c \ \mbox{implies} \int |f_n|^p < c^p$$, then $$\sup_n \int |f_n|^{1 + (p-1)} < \infty \ \mbox{with} \ p-1 >0.$$ Then apply Vitali and the result is immediate !! Thank you. – Both Htob Dec 1 '15 at 12:01
• – user147263 Jan 8 '16 at 7:11