If $f$ is a function such that $f \in L^\infty \cap L^ {p_0}$ where $L^\infty$ is the space of essentially bounded functions and $ 0 < p_0 < \infty$. Show that $ || f|| _{L^p} \to ||f || _{L^\infty} $ as $ p \to \infty$. Where $|| f||_{L^\infty} $ is the least $M \in R$ such that $|f(x)| \le M$ for almost every $x \in X$.
The hint says to use the monotone convergence theorem, but i can't even see any pointwise convergence of functions.
Any help is appreciated.
Answer
Hint: Let $M\lt\|f\|_{L^\infty}$ and consider
$$
\int_{E_M}\left|\frac{f(x)}{M}\right|^p\,\mathrm{d}x
$$
where $E_M=\{x:|f(x)|\gt M\}$. I believe the Monotone Convergence Theorem works here.
Further Hint: $M\lt\|f\|_{L^\infty}$ implies $E_M$ has positive measure. On $E_M$, $\left|\frac{f(x)}{M}\right|^p$ tends to $\infty$ pointwise. MCT says that for some $p$, the integral above exceeds $1$.
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