提出課題2024/11/13

\(f(x)\leq\|{f}\|_{L^{\infty}}\,\text{a.e.}\)より \[\begin{split} \|{f}\|_{L^p} &= \left(\int_{X}\lvert{f(x)}\rvert^p\mathrm{d}{\mu(x)}\right)^{\frac{1}{p}} \\ &\leq\left(\int_{X}\|{f}\|_{L^{\infty}}^{p}\mathrm{d}{\mu(x)}\right)^{\frac{1}{p}} \\ &=\left(\|{f}\|_{L^{\infty}}^{p}\int_{X}\mathrm{d}{\mu(x)}\right)^{\frac{1}{p}} \\ &= \|{f}\|_{L^{\infty}}(\mu(X))^{\frac{1}{p}} \end{split}\] ここで、\(\mu(X)<\infty\)より \[\limsup_{p\to\infty}(\mu(X))^{\frac{1}{p}} = 1\] であるから、 \[\begin{split} \limsup_{p\to\infty}\|{f}\|_{L^p} &\leq \|{f}\|_{L^{\infty}}\limsup_{p\to\infty}(\mu(X))^{\frac{1}{p}} \\ &= \|{f}\|_{L^{\infty}} \end{split}\]