Problem: Let $(\Omega,\mathcal F,\mu)$ be a measure space such that $\mu(A)>0$ forces $\mu(A)\geq1$ for $A\in\mathcal F$. Let $1\leq p<q<\infty$. Then $L^p(\Omega)\subseteq L^q(\Omega)\subseteq L^\infty(\Omega)$ and $\|f\|_\infty\leq\|f\|_q\leq\|f\|_p.$
My Thoughts: Fix $1\leq q<\infty$. Let $0<\delta<\|f\|_\infty$ be given. By the defintion of the essential supremum, it follows that $0<\mu(\{\vert f\vert^q\geq[\|f\|_\infty-\delta]^q\})<\infty$. But by assumption, this forces that $\mu(\{\vert f\vert^p\geq[\|f\|_\infty-\delta]^q\})\geq1$. Put $A_\delta=\{\vert f\vert^q\geq[\|f\|_\infty-\delta]^q\}$. Using the properties of the Lebesgue integral we then obtain $$[\|f\|_\infty-\delta]^q\leq[\|f\|_\infty-\delta]^q\mu(A_\delta)=[\|f\|_\infty-\delta]^q\int_\Omega\mathbf1_{A_\delta}\,d\mu\leq\int_\Omega\vert f\vert^q\,d\mu.$$ Taking $q$th roots we see that $\|f\|_\infty-\delta\leq\|f\|_q$. Since the last inequality holds for all $0<\delta<\|f\|_\infty,$ it follows that $\|f\|_\infty\leq\|f\|_q$.
Now fix $1\leq p<q<\infty$. Since $\|f\|_p$ is assumed to be finite, recalling that $$\|f\|_p^p=p\int_0^\infty t^{p-1}\mu(\{\omega\in\Omega:\vert f(\omega)\vert>t\})\,dt,$$ we see that the hypothesis that $\mu(A)>0$ forces $\mu(A)\geq1$, implies that $\mu(\{\omega\in\Omega:\vert f(\omega)\vert>t\})=0$ for all $t\geq t_0$ for some $t_0>0.$ Therefore, we reduce the problem to proving the following inequality for all $1\leq p<q<\infty,$ $$\left[q\int_0^N x^{q-1}\,dx\right]^{1/q}\leq\left[p\int_0^N x^{p-1}\,dx\right]^{1/p},$$ for fixed $N>0.$ But this is in fact an equality as can be seen from evaluating the integrals. Therefore, the result follows.
Do you agree with my line of reasoning? I am especially distrustful of my argument for the second part of the proof.
Thank you for your time and most valuable feedback.
A measure with the described property is purely atomic, meaning that every set of positive measure does not contain sets of smaller positive measure. Therefore the $L_p$ spaces on such a measure space are esentially $\ell_p$ spaces, for which the inclusions and the inequalities you mentioned are known to be true.