I want to construct a measure space $(X, \mathscr{M}, \mu)$ such that
(a) $\mu(X) = \infty$ (b) For any measurable function $f,$ $f \in L^p$ for some $1 < p < \infty$ implies that $f \in L^1$
(The actual statement is "Construct a measure space such that $\mu(X) = \infty$ and $f \in L^p$ for some $1 < p < \infty$ implies $f \in L^1$". I think the qunatifier at $f$ is for all ?)
I know in general that if the space is not finite measure, no incusion between $L^p$. Besides Lebesgue measure space on $\mathbb{R}^n$, another measure space I can think of is the counting measure space. However, I think I saw the result that $l^p \subset l^q$ for $p \leq q$ (reverse of $L^p$). So I do not think this will work.
Instead of $\mu(X)<\infty$ it's enough if there exists $c<\infty$ such that $\mu(E)\le c$ for every measurable set $E$ such that $\mu(E)<\infty$. So construct a space with an "infinite atom": A set $E$ with $\mu(E)=\infty$ such that $E$ has nno proper measurable subsets.