In Furstenberg-Kesten theorem, a theory relating to products of random matrices, one of the assumptions is that:
$$\log^{+}||A||\in L^1(\mathbb{P}),$$
where $A$ (a random matrix) is the generator of the cocycle. My question is, can anyone explain what is meant by this assumption? In particular, what is meant (or what might be meant) by $L^1(\mathbb{P}).$
Given a probiability space $(\Omega, \mathcal A, \mathbb P)$, by $L^1(\mathbb P)$ one denotes the vector space of (depending on context real or complex) random variables $X \colon \Omega \to \mathbb R$ (or $\mathbb C$) such that $$ E_{\mathbb P}\bigl(\left|X\right|) = \int_\Omega \left|X\right| \, d\mathbb P $$ is finite. As $\log^+ \|A\|$ is non-negative, in your case the assumption just says that the expected value $E(\log^+ \|A\|)$ is finite. See also in this wiki article on $L^p$-spaces.