I am a student who just started to learn basic concepts of ergodic theory.
It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But the books I am reading doesn't really convince me why it is good to have invariant measures.
For example, the Gauss map on the unit interval $x \mapsto \{ 1/x \}$ has the invariant measure $ 1/({1+x})$. What kind of effective results can we prove about the Gauss map using this invariant measure?
One important case is when you have a probability measure, in which case if a map has an invariant measure, then it preserves probability. This is the proper setting for the Birkhoff ergodic theorem, which presumably you will soon learn.