The Kadec-Peł czynski theorem in $L^p$, $1\le p<2$

By a classical result of Kadec and Pełczynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of $\ell^2$ or to the unit vector basis of $\ell^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem is that the limit random measure $μ$ of the sequence satisfies $\int_{\mathbb{R}} x^2 dμ(x)\in L^{p/2}$.

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