If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed measure with some $\mathcal{A}$ a Borel partition of $X$. I have an idea that it means:
\begin{gather} (T_\ast \mu )(A) = \mu(T^{-1} (A)). \end{gather}
Just struggling to understand this definition. Thanks in advance for any insight you can provide.
Also, I only tagged the areas that are related to the field I'm reading this paper in. I'm unsure what the heck it should be tagged as; thanks to any changes that occur here as well if you see a better tag it should be under.
Wow, just to clarify everything, I should have taken some functional analysis at some point. Luiz, thank you, your answer was correct and that was the main question I asked. $T_\ast \mu = \mu(T^{-1})$, and this is commonly called the push through measure. I had completely forgot about this notation since taking some measure theory. But to answer the other part, if one ever sees $\mu(f)$ notation, this simply means Lebesgue integration; i.e. $\mu(f) = \int f \, d \mu$. This is quite common in every functional analysis text I've picked up since when I posted this. Thanks for the help from everyone as usual.