The Kadec-Pełczyński theorem in \(L^p\), \(1\leq p<2\) (Q2790921)

Let \((\Omega, \Sigma, P)\) be a probability space and \(1 \leq p< 2\). A sequence \((X_n)\) in \(L_0 = L_0(P)\) is said to be determining if, for any \(A \in \Sigma\) with \(P(A)> 0\) and all \(t \in \mathbb R\), there exists the limit \(F_A(t) = \lim_{n \to \infty} P(X_n \leq t |A)\), where \(P(X_n \leq t |A)\) is the conditional probability. It is known that, for any determining sequence \((X_n)\), there exists a random measure \(\mu\) such that, for any \(A \in \Sigma\) with \(P(A)> 0\) and any continuity point \(t\) of \(F_A\), one has \(F_A(t) = \mathbb{E}_A (\mu(-\infty,t])\), where \(\mathbb{E}_A\) denotes the conditional expectation with respect to \(A\). The measure \(\mu\) defined above is called the limit random measure of \((X_n)\). The main results asserts that, for any normalized weakly null determining sequence \((X_n)\) in \(L_p\) with uniformly integrable powers \(|X_n|^p\), \(n \in \mathbb N\), the following are equivalent: (i) \((X_n)\) contains a subsequence equivalent to the unit vector basis of \(\ell_2\); (ii) \(\int_{-\infty}^\infty x^2 \, d \mu(x) \in L^{p/2}\). As noted by the authors, the determining assumption on \((X_n)\) is not actually restrictive because any sequence \((X_n)\) with uniformly integrable powers \(|X_n|^p\), \(n \in \mathbb N\), contains a determining subsequence.