I am trying to read a somewhat "old" paper by Daniel Rudolph: "x2 x3 invariant measures and entropy". I have some problems trying to understand Rudolph's notation for conditional expectations. More specifically in page 402 he writes $$\mathbb{E}_{\hat{\mu}}(T^n(\varphi^{-1}(x_1(\hat{y}) + \frac{t}{p})) \ | \ \varphi^{-1}(x_1(\hat{y}))) := \delta(\hat{y},t,n)$$ The probability spaces is $\hat{Y}$ with point $\hat{y} \in \hat{Y}$.
Anyone understands how to traduce that conditional expectation to more modern notation? I'm used to seeing the clasical $\mathbb{E}(f \ | \ \mathcal{F})(x)$ notation. I did not want to specify what each of the functions $\varphi, x_1, T$ as I'm really looking for a general undestanding of the notation.
Besides if anyone knows the paper and is willing to explain not only the notation but the intuitive meaning of the conditional expectation shown before it would be very welcomed.
Thanks