Let be a space$X$, $\mu:S\rightarrow[0,+\infty]$ a positive measure, with $S$ a $\sigma-algebra$ of sets on $X$ and $L_p(\mu)=\{f:\mathbb{R}\rightarrow \mathbb{K}\ | \int_{X}|f|^pd\mu<+\infty\}$. Let be $1\leq p\leq q < +\infty$, prove that: $$a)\ \inf\{\mu(E):E\in S \land \mu(E)>0\}=0 \Longrightarrow L_p(\mu) \setminus L_q(\mu) \neq \emptyset$$ $$b)\ \sup\{\mu(E):E\in S \land \mu(E)<+\infty\}=+\infty \Longrightarrow L_q(\mu) \setminus L_p(\mu) \neq \emptyset$$
So, for the point $(a)$ I was able to create a disjoint sequence of sets $E_n$ which $0<\mu(E_n)<\frac12^n$ for every $n\geq1$ and I wanted to create a function $u$ like this:$$u(x)=\sum_{n\geq1}a_n1_{E_n}(x)$$ such that $u \in L_p(\mu)$ but $u \notin L_q(\mu)$, but I struggle to find such $a_n$. Any help? A note: I define the space $L_p(\mu)$ as such: $$L_p(\mu)=\{f:X\rightarrow\mathbb{K} \ | \int_X|f|^pd\mu<+\infty\}$$ where every element of $L_p(\mu)$ is a equivalence class [f], denoted as $f$ to simplify, defined as: $$[f]=\{g:X\rightarrow\mathbb{K} \ | \int_X|f-g|^pd\mu=0\}$$
Since the sets $E_n$ are pairwise disjoint, we can see that $$ \int \lvert u\rvert^pd\mu=\sum_{n\geqslant 1}\lvert a_n\rvert^pc_n, $$ where $c_n=\mu(E_n)$ (of course, this equality also holds replacing $p$ by $q$ on both sides). Let $a_n=\left(n^2c_n\right)^{-1/p}$. With this choice, $a_n$ belongs to $L^p$, but not to $\mathbb L^q$, as $$ \sum_{n\geqslant 1}\lvert a_n\rvert^qc_n=\sum_{n\geqslant 1}n^{-2q/p}c_n^{1-q/p} $$ and since $q>p$, $c_n^{1-q/p}\geqslant 2^{n(q/p-1)}$.
For point (b), we can use a similar construction, but this time based on sets $E_n\in\mathcal S$ such that $n<\mu(E_n)<\infty$.