Suppose $f \in L^0$. I read that for a general measure space, if $\mu(X)<\infty$, then we cannot have that both $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$, but if $\mu(X)=\infty$, then we can find an $f$ such that this statement is true. I'm not sure how to show this for the finite case and am having trouble thinking of an example of a function $f$ for the infinite case such that $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$.
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What is $L^0$? Is it $L^\infty$? – Nov 13 '14 at 02:51
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$L^0={f:f$ is a measurable function from $X$ to a field $\mathbb{k}}$ – user192119 Nov 13 '14 at 02:59
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Hint:
For the first case, you can actually show that
$$||f||_p \leq ||f||_\infty \mu(X)^{1/p}$$
if $\mu(X) < \infty$. For the second one, consider $X = \mathbb R$ with the Lebesgue measure. Can you found a bounded positive function $f$ on $\mathbb R$ so that
$$\int_\mathbb R f dx = + \infty?$$
(Don't think too hard)
KCd
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1Yes, for example, let $f\equiv 1$, then? :) (Don't think too hard) @John I like this sentence. – spatially Nov 13 '14 at 03:22
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Oh, that makes sense! So for a general measure space, if $\mu(X)=\infty$, we could just make $f$ the characteristic function of $X$? – user192119 Nov 15 '14 at 20:05
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