Let $\Omega \subset \mathbb{R}^d$ and $f \in L^\infty(\Omega)$. We know that if $\lambda(\Omega) < +\infty$ with $\lambda$ the Lebesgue measure on $\mathbb{R}^d$, we have the inclusion
$$L^\infty(\Omega) \subset L^p(\Omega), \quad \forall p \in [1,+\infty[.$$ and that this injection is continuous.
I am looking for references or a proof for the following property : $||f||_{L^\infty(\Omega)} = \underset{p \rightarrow + \infty}{\liminf} ||f||_{L^p(\Omega)}.$
Any help is welcomed.
Assume $f\in L^{\infty}$. Then, $|f|\leq \|f\|_{\infty}$ almost everywhere and we get that
$$ \|f\|^p_p=\int |f|^p\textrm{d}\lambda\leq \int \|f\|_{\infty}^p\textrm{d}\lambda=\lambda(\Omega)\|f\|^{p}_{\infty}, $$ implying that $\limsup_{p\to\infty}\|f\|_p\leq \|f\|_{\infty}$.
Similarly, for any $\|f\|_{\infty}>\varepsilon>0$, we have that $|f|\geq 1_{\{|f|\geq \|f\|_{\infty}-\varepsilon\}} (\|f\|_{\infty}-\varepsilon)$ and we get that $$ \|f\|_p\geq \lambda(\{|f|\geq \|f\|_{\infty}-\varepsilon\})^{1/p}(\|f\|_{\infty}-\varepsilon) $$ As $\lambda(\{|f|\geq \|f\|_{\infty}-\varepsilon\})>0,$ we get that $$ \liminf_{p\to\infty}\|f\|_p\geq \|f\|_{\infty}-\varepsilon, $$ Since $\varepsilon>0$ was arbitrary, we get the result.