Let \((f_n)\) be real-valued functions in \(L^{4/3}(0,1)\) such that \(f_n\to0\) in measure and \[ \int_0^1 |f_n(x)|^{4/3}\,dx\le1 \qquad(n\ge1). \] Show that \[ \int_0^1|f_n(x)|\,dx\longrightarrow0. \]
Solution
Fix \(\varepsilon>0\). Choose \(\eta=\varepsilon/2\) and split \[ \int_0^1|f_n| = \int_{\{|f_n|\le\eta\}}|f_n| + \int_{\{|f_n|>\eta\}}|f_n|. \] The first term satisfies \[ \int_{\{|f_n|\le\eta\}}|f_n|\le\eta=\frac\varepsilon2. \] For the second term, Hölder’s inequality with exponents \(4/3\) and \(4\) gives \[ \begin{aligned} \int_{\{|f_n|>\eta\}}|f_n| &\le \left(\int_0^1|f_n|^{4/3}\right)^{3/4} m\{|f_n|>\eta\}^{1/4}\\ &\le m\{|f_n|>\eta\}^{1/4}. \end{aligned} \] Because \(f_n\to0\) in measure, \[ m\{|f_n|>\eta\}\longrightarrow0. \] Hence for all sufficiently large \(n\), \[ m\{|f_n|>\eta\}^{1/4}<\frac\varepsilon2. \] Therefore \[ \int_0^1|f_n|<\varepsilon \] for all sufficiently large \(n\), proving \(f_n\to0\) in \(L^1(0,1)\).