If $\phi : X \rightarrow \mathbb{R}$ is a measurable function and $\mu$ is a measure on $X$ then what is $``\mu(\phi)"$?
Is this a notation for some function?
I came across this notation for the first time in Theorem 1.2 (page 2) stated in these lectures, http://www.hairer.org/papers/harris.pdf
It is simply the integral of $\phi$ with respect to the measure $\mu$, that is $\mu(\phi) := \int_X \phi(x) \mu(dx)$
Hence, you can see the measure $\mu$ as a linear function from the set of measurable functions on $X$ to $[0,\infty]$