  For linear random dynamical systems in a separable Banach space $X$, we
derived a series of Krein-Rutman type Theorems with respect to co-invariant
cone family with rank-$k$, which present a (quasi)-equivalence relation between
the measurably co-invariant cone family and the measurably dominated splitting
of $X$. Moreover, such (quasi)-equivalence relation turns out to be an
equivalence relation whenever (i) $k=1$; or (ii) in the frame of the
Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater
than the negative infinity.
  For the second case, we thoroughly investigated the relations between the
Lyapunov exponents, the co-invariant cone family and the measurably dominated
splitting for linear random dynamical systems in $X$.
