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The well-known Conley's theorem states that the complement of chain recurrentset equals the union of all connecting orbits of the flow $\phi$ on the compactmetric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where$\mathcal{CR}(\phi)$ denotes the chain recurrent set of $\phi$, $A$ stands foran attractor and $B(A)$ is the basin determined by $A$. In this paper we showthat by appropriately selecting the definition of random attractor, in fact wedefine a random local attractor to be the $\omega$-limit set of some randompre-attractor surrounding it, and by considering appropriate measurability, infact we also consider the universal $\sigma$-algebra $\mathcalF^u$-measurability besides $\mathcal F$-measurability, we are able to obtainthe random case of Conley's theorem.Comment: 15 page

The random case of Conley's theorem