Why don't we write the error term with forecasts? A minor issue that's been annyong me as I blindly write it...
As an example for a simple AR(1) process
$$y_{t} = Ay_{t-1} + \varepsilon_{t}$$
I can write the process at time t+1:
$$y_{T+1} = Ay_{T} + \varepsilon_{T+1}$$
But when I project on time t information, I need to write it like so:
$$y_{T+1,T} = Ay_{T} $$
Obviously, this works for when I need to compute the forecast error:
$$ y_{T+h} - y_{T+h,T}$$
But I don't actually know the rationalè for doing this. Oddly my lecture notes do not mention why, and a quick google to find the intiution beind this reveals very little.
Is the intuition simply that we don't write the residuals when taking our actual results, but we do represent the errors on a theoretical forecast?
 A: The point is that the realizations of your time series are stochastic. In your example, by means of autoregressive modelling (AR) you build a model on your data that allows to make a prediction. Of course, your model is not likely to predict exactly the value of your time series at a given time, but you will make a prediction error.
Notice that before building your model, you make some assumptions on the error structure of your predictions, which indeed are expected values at some given instants.
In other words, when you build a model like the one you mentioned, you take a stochastic process and split it into two parts: a deterministic one and a stochastic one. Then, you make your predictions by considering the deterministic part, and the prediction errors represent the stochastic part of the process (you can't predict the errors, but you make an assumption on their structure --- e.g. they are normally distributed with mean zero and variance $\sigma^{2}$). 
For example, if you consider a stochastic process $X_{t}$, you can build an AR(2) model such as:
$$ x_{t} = x_{t-1}\phi_{1} + x_{t-2}\phi_{2} + \epsilon_{t}.$$
In such a model, $x_{t-1}\phi_{1} + x_{t-2}\phi_{2}$ represents the deterministic part, whereas $\epsilon_{t}$ is the prediction error, i.e. the stochastic part.
When you have to make a prediction you use only the deterministic part, i.e. in order to predict $x_{t}$ you substitute to $x_{t-1}$ and $x_{t-2}$ the past realizations of your process and to $\phi_{1}$ and $\phi_{2}$ the estimated coefficients. You don't consider the stochastic part, $\epsilon_{t}$, because you have assumed that it is normally distributed with zero mean and fixed variance, so that its expected value is zero.  
A: We are given that
$$y_{t} = Ay_{t-1}+\varepsilon_t$$ 
where $\operatorname{E}(\varepsilon_t)=0$ for each $t$.
Also, $\operatorname{E}(\varepsilon_{t+h}|I_t)=0$ for $h>0$, for each $t$, where $I_t$ is information available at time $t$.
We define
$$y_{T+1,T} := \operatorname{E}(y_{T+1}|I_T).$$ 
Thus $y_{T+1,T}$ is the conditional mean of $y_{T+1}$ given information available at time $T$. Once this definition is explicit, there is no need to worry about a "missing" forecast error as can be seen below. We can expand and substitute using the two equations above to make things more obvious: 
$$\operatorname{E}(y_{T+1}|I_T) = \operatorname{E}(Ay_T+\varepsilon_{T+1}|I_T) = \operatorname{E}(Ay_T|I_T)+\operatorname{E}(\varepsilon_{T+1}|I_T) = Ay_T+0=Ay_T$$
So there is nothing missing in writing $y_{T+1,T}=Ay_T$.
To answer the title question, 

Why don't we write the error term with forecasts?

it is because when forecasting the conditional mean, the error term vanishes as conditional mean is additive and the erorr's conditional mean is zero. If, on the other hand, we were forecasting something else than the conditional mean, the error term may be relevant and could be included in the forecast (e.g. forecast from a GARCH model: conditional variance of the error term is not zero and appears in the forecast).
