Let $(X,\mathcal M,\mu)$ be a measure space such that $ \mu(X)<\infty$. Suppose that $(f_n)$ is a sequence in $L^p(X)$ such that $|f_k|$ converges weakly to $|f|$ in $L^p(X)$. A solution to a problem I found in a book (Problems and Solutions in Mathematics – Major American Universities PHD Qualifying Questions and Solutions) says that it follows by the Vitali-Hahn-Saks for any $ \varepsilon >0$ there exists $ \delta>0$ such that $$ \int_E |f_k| d\mu + \int_E |f| d \mu < \varepsilon$$
whenever $\mu(E)<\delta$.
My question is:
What is the Vitali Hahn Saks Theorem (I remember seeing it, but I forgot where, and Google didn't help much) and how does the theorem apply here.
Best Answer
The version of Vitali-Hahn-Saks they are using is found e.g. in Yosida, "Functional Analysis", sec. II.2. The measures $|f_k| \ d\mu$ and $|f| \ d\mu$ are absolutely continuous wrt $\mu$, and the theorem asserts that this absolute continuity is uniform in $\mu$.