$\mathcal{L}^{p}(\mu) \backslash \mathcal{L}^{p'}(\mu) \neq \emptyset$

Question

Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $1 \leq p' < p \leq \infty$ and $\mu$ be $\sigma$-finite but not finite. I am trying to prove that $\mathcal{L}^{p}(\mu) \backslash \mathcal{L}^{p'}(\mu) \neq \emptyset$. Hence I am trying to find an $f \in \mathcal{L}^{p}(\mu)$ such that $f \not \in \mathcal{L}^{p'}(\mu)$. Since $\mu$ is $\sigma$-finite, I thought I would start by decomposing $\Omega = \bigcup_{k = 1}^\infty A_k$, where $A_k \in \mathcal{F}$ and $\mu (A_k) < \infty$ for all $k \in \mathbb{N}$. Then I thought about setting $f = \sum_{k = 1}^\infty a_k 1_{A_k}$ for suitably defined $a_k$ such that the function has finite $p$-norm but infinite $p'$-norm. However, I am having troubles finding such $a_k$. Any ideas? Thanks in advance!

Note: $\mathcal{L}^p (\mu)$ is the set of real valued measurable functions with finite $p$-norm.

In my question $p > 1$, so this example does not really help me here. Also, my measure space is given. — Wittgenstein's Poker, Sep 25 '22 at 14:22
See DavideGiraudo's posting. That should answer your question. — Mittens, Sep 25 '22 at 15:48
Where exactly does he provide an example of an $f$ that is in $\mathcal{L}^p(\mu)$ but not in $\mathcal{L}^{p'}(\mu)$? I don't see it. Note that my measure space is assumed to be $\sigma$-finite, but not finite. — Wittgenstein's Poker, Sep 25 '22 at 15:57
Ah, I didn't even think to negate the statement of the theorem. That works, thanks a lot! — Wittgenstein's Poker, Sep 25 '22 at 16:43
I just realize that there is no need to restrict $p$ and $p'$ to $[0,\infty)$. In fact, we can consider any $0<p<p'<\infty$. I posted this observation here — Mittens, Sep 30 '22 at 17:10

$\mathcal{L}^{p}(\mu) \backslash \mathcal{L}^{p'}(\mu) \neq \emptyset$

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