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Let $X= [0,1), S= \mathcal{B}_{[0,1)}, \lambda = $Lebesgue measure in $\mathcal{B}_{[0,1)} $ Prove

(a) $L_p(\lambda) \subsetneq \bigcap_{0<r<p} L_r(\lambda) $ for every fixed p.

(b) $L_{\infty}(\lambda) \subsetneq \bigcap_{0<p<+\infty} L_p(\lambda) $

And let $X= [0,+\infty), S= \mathcal{B}_{[0,+\infty)}, \lambda = $Lebesgue measure in $\mathcal{B}_{[0,+\infty)} $. Let $p \in [0,+\infty) $ be fixed. Prove $\bigcup _{p<q}L_q(\lambda) \subsetneq L_p(\lambda) $.

Can someone give me a hint?

Thanks in advance.

Deni
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1 Answers1

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(a) For the inclusion, use and show $|f(x)|^r\leqslant |f(x)|^p +1$. For strictness: $x^{-1/p}$.  

(b) If $f$ belongs to $L^\infty$, $f$ is almost surely bounded by a constant, hence so are all the powers of $f$. Constants on finite measure spaces are integrable. For strictness: $f(x)=\log x$.

Davide Giraudo
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