Let $(\Omega,R)$ denote a flow, where $\Omega$ is compact metric space and $\gamma$ is a normalized measure on $\Omega$.
I have problems to understand the following passage in http://www.sciencedirect.com/science/article/pii/0022039678900773, page 26, Line (5):
Since the flow $(\Omega,R)$ is uniquely ergodic, one has, for each $\omega_0\in\Omega$, $$ \lim_{\lvert t\rvert\to\infty}\frac{1}{t}\int_0^ta(\omega_0\cdot s)\, ds=\int_{\Omega}a(\omega)\, d\gamma(\omega)=\text{const}\equiv a. $$
The first equation is just a property that follows from the ergodicity of the flow.
But I do not understandwhy the constant is equal to $a$.
Or is it just a not so good notation to call the constant $a$ (because the function in the integrand is already called $a$?)
With other words: Are the function in the integrand and the constant the same?