$\forall f\in L^p : f\in L^1$

Question

Yesterday I asked this question.

exists $f\not\in L^1(X)$ such that $f\in L^p(X)$

Which I already managed to solve, however that result generated me a doubt.

My question

$\exists (X,M,\mu)$ measurement space $\wedge$ $\exists p(1,\infty):\forall f\in L^p(X):\mu(X)=\infty$ $\wedge$ $f\in L^l(X)$

Under these conditions can such $X$ and $p$ exist?

Update

There will exist a space of Measurement X such that $\mu(X)=\infty$, and there will exist a $p(1,\infty)$ such that $L^p(X) \subseteq L^1(X)$