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given p>1, whats an example of f where $\int_{-\infty}^{\infty} |f| < \infty$ but $\int_{-\infty}^{\infty} |f|^p = \infty$

what about vica-versa? that is whats an example of g where $\int_{-\infty}^{\infty} |g| = \infty$ but $\int_{-\infty}^{\infty} |g|^p < \infty$

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up vote 7 down vote accepted

The former is caused by functions that blow up too rapidly near a point. Think about functions of the form $f(x) = x^{-a} 1_{[0,1]}$.

The latter is caused by functions that decay too slowly near $\infty$. Think about functions of the form $f(x) = x^{-a} 1_{[1,\infty)}$.

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ok for the first one i came up with x^(-1/p). for the second one i have x^(-1). –  jack Feb 11 '11 at 15:40
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