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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 $L^p(X) \subseteq L^1(X)$

F.R.
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    English is probably a better medium of expression than propositional logic. – copper.hat Nov 21 '22 at 16:36
  • If I understand correctly, you’re asking for a measure space $X$ with infinite measure and a $p \in (1,\infty)$ such that $L^p(X)$ is a subset of $L^1(X)$? – kieransquared Nov 21 '22 at 16:40
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    In this case, define a measure $\mu$ on $\mathbb{R}$ with $\mu(A)=\infty$ whenever $A$ is nonempty. Then the only integrable function is the zero function, and hence $L^1 = L^p = {0}$ for every $p$. – kieransquared Nov 21 '22 at 16:49
  • Look at https://math.stackexchange.com/a/1529073/27978, and extend the function to be zero on $(0,1)^c$. – copper.hat Nov 21 '22 at 16:53
  • Whoa, that is a big change in question. – copper.hat Nov 21 '22 at 16:53
  • This may be duplicate of this posting https://math.stackexchange.com/q/66029/121671 – Mittens Mar 08 '25 at 00:28

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