Usefulness of invariant measures with full measure on some set I am reading Arnold's book Random Dynamical Systems and he makes the following statement.
"in some cases it is useful to construct a $\phi$ invariant measure $\mu$ such that $\mu(A)=1$ for some $A\in \mathcal{F}\otimes \mathcal{B}$."
Here $\mathcal{F}$ is the $\sigma$-algebra on our probability space and $\mathcal{B}$ is the $\sigma$-algebra along each fiber. This may come later in the readings but I was interested in why constructing such measures is useful? Can such measures be used to study trajectories within some invariant set $A$ since $\mu(\phi^{-1}(A))=\mu(A)=1$? Any comments would be much appreciated.
 A: Sinai, in his 1976 notes Introduction to Ergodic Theory lists the existence (hence also construction) of an invariant measure for a system to be one of the statistical properties one would be interested in. There are many benefits to invariant measures in general; here are some (all roughly stated):

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*An invariant measure is roughly a notion of a space average; whence if there is some kind of an ergodic theorem available that means that one is able to take time averages also.


*Any measure $\mu$ allows one to consider the space of  $p$-integrable observables $L^p(\mu)$ on the state space, and any such space carries a system intimately related to the original system, called the Koopman system (see Mathematical framework of the Koopman operator). If the measure $\mu$ is invariant under the original dynamics, then the Koopman system often becomes an isometry.


*If in addition the invariant measure is ergodic (often a decomposition of an invariant measure into ergodic components is available), then one can consider Lyapunov exponents of the system not as functions on the phase space but indeed as numbers (see Why do we refer to Lyapunov exponents as a characteristic of the system, telling us about its chaoticity, when it is only referred to a point?).


*The support of an invariant measure is often intimately connected to the periodic points of the system. More generally, the regions where an invariant measure is concentrated is also indicative of high degree of recurrence, hence complexity.
I should note that the discussions at the links are for the deterministic case, although there are versions of these statements for the random case also.
For your specific condition, the fibers may have infinite measure; still your condition requires to be able to single out a certain region of the total space to act as a probability space (note that the base is already a probability space). Thus in a sense this is a probabilistic analog of a factor map admitting a fundamental domain.
